CFD BASED AIRFOIL SHAPE OPTIMIZATION FOR AERODYNAMIC DRAG

REDUCTION

by

Mohammed Taha Shafiq Khot

A Thesis Presented to the Faculty of the American University of Sharjah

College of Engineering

in Partial Fulfillment of

the Requirements for the Degree of

Master of Science in

Mechanical Engineering

Sharjah, United Arab Emirates

May 2012

© 2012 Mohammed Taha Shafiq Khot. All rights reserved

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Abstract

Commercial airplanes generally follow specific flight profiles consisting of

take-off, climb, cruise, descend and landing. These flight profiles essentially change

the freestream conditions in which the aircrafts operate. Furthermore, over the course

of the flight, the required lift force changes as the fuel gets consumed. The

conventional fixed wing designs account for these requirements by catering to

multiple but fixed design points which, however, compromise the overall flight

performance. Employing adaptive wing technology allows to fully exploring the

aerodynamic flow potential at each point of the flight envelope. The objective of this

research is to develop an aerodynamic optimization framework for optimizing a

baseline airfoil shape at specific off-design operating points within a typical transonic

flight envelope. The objective function is the lift-moment constrained drag

minimization problem. B-spline curve fitting is used to parameterize the airfoil

geometry and the control points along with the angle of attack are used as the design

variables for the optimization process. An iterative response surface optimization

methodology is employed for carrying out the shape optimization process. Design of

Experiments (DoE) using the Latin Hypercube Sampling algorithm is used to

construct the response surface model. This model is then optimized using the SQP

technique. The various parameters that gauge the aerodynamic performance (lift, drag

and moment coefficients) are obtained using CFD simulations. GridPro is used as the

meshing tool to generate the flow mesh, and the CFD simulation is performed using

ANSYS-FLUENT. The parameterization, design of experiments, and the response

surface model optimization are performed using MATLAB. RAE 2822 design study

is carried out to validate the optimization algorithm developed. The adaptive airfoil

concept is demonstrated using a Boeing-737 classic airfoil at three steady flight

operating points that lie within a typical aircraft flight envelope. The aerodynamic

performance of the adaptive airfoil is then compared to that of the baseline airfoil.

Search Terms—Aerodynamics, CFD, optimization, design of experiments, response

surface modeling.

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Table of Contents

Abstract .......................................................................................................................... 4

List of Figures ................................................................................................................ 9

List of Tables ............................................................................................................... 11

Nomenclature ............................................................................................................... 12

Acknowledgements ...................................................................................................... 13

1. INTRODUCTION ................................................................................................... 15

1.1 Motivation .......................................................................................................... 16

1.2 Historical perspective ......................................................................................... 17

1.3 Problem Statement ............................................................................................. 22

1.4 Review of Literature........................................................................................... 24

1.5 Thesis Objectives ............................................................................................... 26

2. OPTIMIZATION AND CFD .................................................................................. 28

2.1 Introduction to Optimization .............................................................................. 28

Mathematical Formulation .......................................................................... 29 2.1.1

Characteristics of Optimization Algorithms ............................................... 29 2.1.2

2.2 Sequential Quadratic Programming ................................................................... 30

2.2.1 The SQP Technique .................................................................................... 30

2.2.2 Solution Procedure ...................................................................................... 31

Updating the Hessian Matrix ...................................................................... 31 2.2.3

2.3 Design Variables ................................................................................................ 32

B-Spline Curves Introduction ..................................................................... 32 2.3.1

B-Spline Formulation.................................................................................. 33 2.3.2

Airfoil parameterization using B-Spline curves ......................................... 33 2.3.3

Design Variables Vector ............................................................................. 38 2.3.4

2.4 Objective Functions............................................................................................ 38

Inverse design ............................................................................................. 38 2.4.1

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Maximizing Aerodynamic Efficiency......................................................... 39 2.4.2

Lift-Constrained Drag Minimization .......................................................... 39 2.4.3

2.5 Flow Equations ................................................................................................... 40

The Mass Conservation Equation ............................................................... 41 2.5.1

Momentum Conservation Equations........................................................... 42 2.5.2

Viscous Energy Equation ............................................................................ 42 2.5.3

Turbulence Model ....................................................................................... 43 2.5.4

2.6 Computational Fluid Dynamics ......................................................................... 48

Discretization .............................................................................................. 49 2.6.1

CFD based Shape Optimization .................................................................. 51 2.6.2

3. ITERATIVE RESPONSE SURFACE BASED OPTIMIZATION ........................ 52

3.1 Introduction ........................................................................................................ 52

3.2 Response Surface Optimization ......................................................................... 53

Design Space ............................................................................................... 55 3.2.1

Design of Experiments ................................................................................ 56 3.2.2

Constructing the RSM ................................................................................ 61 3.2.3

Optimizing the RSM ................................................................................... 65 3.2.4

System Response ........................................................................................ 65 3.2.5

3.3 Iterative Improvement of the RSM .................................................................... 66

Algorithm Flow Chart ................................................................................. 67 3.3.1

3.4 MATLAB Implementation of the Iterative RSO ............................................... 70

MATLAB Optimization Toolbox ............................................................... 74 3.4.1

MATLAB Global Optimization Toolbox ................................................... 77 3.4.2

3.5 Meshing using GridPro ...................................................................................... 79

Surface Specifications ................................................................................. 79 3.5.1

Block Topology .......................................................................................... 81 3.5.2

Run schedule ............................................................................................... 84 3.5.3

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Wall Clustering ........................................................................................... 84 3.5.4

3.6 CFD Solver—ANSYS FLUENT ....................................................................... 86

Pressure-Based Solver ................................................................................ 87 3.6.1

General Scalar Transport Equation ............................................................. 87 3.6.2

Spatial Discretization .................................................................................. 89 3.6.3

Computing Forces and Moments ................................................................ 90 3.6.4

ANSYS FLUENT Setup and Run............................................................... 91 3.6.5

3.6.6 Flow Solver Validation ............................................................................... 93

3.7 Simulation Time ................................................................................................. 94

4. RESULTS AND DISCUSSIONS ........................................................................... 96

4.1 Overview ............................................................................................................ 96

4.2 Case Study#1: Optimization of the RAE 2822 Airfoil ...................................... 96

4.3 Adaptive Airfoil Design based on the Boeing-737C airfoil ............................. 101

4.3.1 Operational Envelope ................................................................................ 102

CL requirements across the envelope ........................................................ 103 4.3.2

Airfoil Geometry ....................................................................................... 104 4.3.3

4.3.4 CASE-A: Mach 0.74 at 9083.3 [m] (29,800[ft]) ...................................... 105

4.3.5 CASE-B: Mach 0.65 at 9083.3 [m] (29,800[ft]) ....................................... 109

4.3.6 CASE-C: Mach 0.65 at 6827.0 [m] (22,400[ft]) ....................................... 113

4.4 Performance Comparison of Baseline Airfoil Relative to Optimized Airfoil .. 117

4.5 Adaptive Airfoils .............................................................................................. 118

5. CONCLUSION AND FUTURE WORK .............................................................. 120

5.1 Summary and Conclusion ................................................................................ 120

5.2 Future Work ..................................................................................................... 123

REFERENCES .......................................................................................................... 125

Appendix A GridPro Topology Input Language (TIL) File (FINAL3.FRA) ............ 128

Appendix B ANSYS-FLUENT JOURNAL FILE (FLUENT_RUN.JOU) ............... 136

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VITA .......................................................................................................................... 138

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List of Figures

Figure 1 AFTI-F-111 Mission Adaptive Wing (MAW) in Flight ............................... 18

Figure 2 Optimum wing cambers at different flight conditions .................................. 19

Figure 3 Variable leading and trailing edge wing design ............................................ 19

Figure 4 Lewis et al variable camber wing command system ..................................... 20

Figure 5 Artist’s rendering of NASA’s Morphing Airplane ........................................ 21

Figure 6 Reparameterization ........................................................................................ 35

Figure 7 B-Spline parameterization of RAE 2822 airfoil ............................................ 37

Figure 8 Continuous Domain and Discrete Domain .................................................... 49

Figure 9 The Computational Spatial Grid .................................................................... 49

Figure 10 Typical finite volume cell ............................................................................ 50

Figure 11 General RSO procedure ............................................................................... 55

Figure 12 Three-variable, ten-point Latin hypercube sampling plan shown in three

dimensions (top left), along with its two-dimensional projections. ........... 58

Figure 13 Lower Limit, Upper limit and Baseline Control Points and Airfoil shapes 60

Figure 14 Iterative RSO process flow chart ................................................................. 69

Figure 15 MATLAB implementation of the Iterative RSO method ............................ 73

Figure 16 GridPro surfaces .......................................................................................... 80

Figure 17 Topology layout for the airfoil and the far-field ......................................... 82

Figure 18 Topology layout in the wake region of the airfoil ....................................... 83

Figure 19 Topology around the airfoil surface ............................................................ 83

Figure 20 Mesh—full display ...................................................................................... 85

Figure 21 Mesh—full and close up view ..................................................................... 85

Figure 22 CFD verification results .............................................................................. 94

Figure 23 B-spline parameterization of the RAE 2822 airfoil ..................................... 97

Figure 24 Range of the B-spline control points (Design variables) ............................. 97

Figure 25 RAE 2822 Baseline and Optimized airfoil shapes CP distributions .......... 100

Figure 26 Plot comparing the actual and predicted function values .......................... 101

Figure 27 Typical aircraft flight envelope ................................................................. 102

Figure 28 Discretized points in the flight envelope. .................................................. 103

Figure 29 Boeing 737 airfoil (baseline) ..................................................................... 104

Figure 30 B-Spline parameterization of the Boeing 737 airfoil (expanded) ............. 104

Figure 31 Range of the Boeing-737 airfoil design variables ..................................... 105

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Figure 32 CASE-A Baseline and Optimized airfoil shapes and their corresponding

surface CP distributions ............................................................................ 108

Figure 33 Plot comparing the actual and the predicted objective function values

(CASE-A) ................................................................................................ 109

Figure 34 CASE-B Baseline and Optimized airfoil shapes and their corresponding

surface CP distributions ............................................................................ 112

Figure 35 Plot comparing the actual and the predicted objective function values

(CASE-B) ................................................................................................. 113

Figure 36 Plot comparing the actual and the predicted objective function values

(CASE-C) ................................................................................................. 116

Figure 37 Plot comparing the actual and the predicted objective function values

(CASE-C) ................................................................................................. 117

Figure 38 Baseline and Optimized Airfoil shapes for Cases A, B, and C ................. 119

Figure 39 Baseline and Optimized Airfoil shapes for Cases A, B, and C (Expanded

View)........................................................................................................ 119

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List of Tables

Table 1 Model Constants ............................................................................................. 48

Table 2 Baseline and Upper and Lower limit Values of the Design Variables ........... 98

Table 3 Freestream conditions for the design operating condition .............................. 98

Table 4 Baseline and Optimized performance at off-design operating point .............. 99

Table 5 Baseline and Optimized values of the Design Variables .............................. 100

Table 6 Baseline and Upper and Lower limit Values of the Design Variables ......... 105

Table 7 Freestream conditions for the off-design operating point A ......................... 106

Table 8 Baseline and Optimized values of the Design Variables .............................. 106

Table 9 Baseline and Optimized performance at off-design operating point ............ 107

Table 10 Freestream conditions for the off-design operating point B ....................... 110

Table 11 Baseline and Optimized values of the Design Variables ............................ 110

Table 12 Baseline and Optimized performance at off-design operating point B ...... 111

Table 13 Freestream conditions for the off-design operating point C ....................... 114

Table 14 Baseline and Optimized values of the Design Variables C ........................ 114

Table 15 Baseline and Optimized performance at off-design operating point C ...... 115

Table 16 Performance Comparison Analysis ............................................................ 118

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Nomenclature

f – Objective function

𝑔𝑖 − Constraints

X —Vector of design variables

L — Langragian formulation

𝜆 — Langrange Multipliers

H — Hessian of the Langrange function

𝑡𝑗 — B-Spline curve Parameter

𝑃𝑖𝑥,𝑦 — B-Spline Control point coordinates

𝑈𝑖 — B-Spline Knot Vector

𝐶𝐿 ,𝐶𝐷 ,𝐶𝑀 — Lift, moment and drag coefficients.

𝐽 — Aerodynamic optimization objective function

CFD — Computational Fluid Dynamics

𝐕 — Flow velocity vector

𝜌 — Fluid density

𝜏𝑖𝑗 — Reynolds Stresses

𝜈 — Kinematic Viscosity

𝐗𝐷𝐷𝐷 — Design of Experiments plan

𝐲 = 𝑓(𝐱,𝜷) — Response Surface Model

𝜷 — Polynomial regression coefficients

𝑛𝑡 — Number of regression coefficients.

𝚽 — Vandermonde matrix of design variables

𝑁𝑠,𝑁var — Number of samples and variables

σ𝑣 — Adjusted Root Mean Square value

𝑅𝑣𝑑𝑗2 — Coefficient of multiple determination

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Acknowledgements

I would like to begin with thanking Almighty God for being my guidance of

the journey of life. He is the Creator and Sustainer of the all perfect nature which has

been a major source of inspiration for many of the engineering solutions that exist

today.

I would like to extend my sincerest thanks and appreciation to Dr. Ali Jhemi.

It has been an honor and a privilege to be his student. He is a great mentor whose

expert opinions and guidance have been central to the successful completion of this

research. Working with him has been both, an enjoyable as well as a great learning

experience. Throughout the duration of my research work, Dr. Jhemi has been patient

and cordial, which encouraged me to work even harder and with greater

determination.

I would especially like to thank Dr. Essam Wahba for helping me understand

and clarifying a lot of doubts with respect to CFD. His expert comments and opinions

on my simulation results have helped me save a lot of time and do things the right

way with confidence. Also I would like to thank Dr. Mohammed Amin Al Jarrah,

head of the Mechanical Engineering department, for being a member of my graduate

thesis committee and gauging my research work. His experience in the area of

aerodynamics has been valuable in defining the scope of my research.

I am very grateful to Dr. Armando Vavalle of Rolls Royce Plc, for his expert

opinions on my results which were crucial for my research. He has been very

forthcoming and helpful, and his recommendations have been very valuable in

making my research work even better. A special appreciation and thanks to Mr.

Samuel Ebenezer of GridPro, for not only providing me with the software but even

tons of information and guidance relating to it.

Finally, I would like to express my deepest gratitude to everyone in my

family, especially my loving parents who have been very kind and understanding

throughout. I owe a lot of respect to them for their fabulous upbringing, taking great

care of me all the while, supporting and motivating me to work harder to get to this

point in life. And so, I dedicate this research to them. Also, I would like to thank all

my friends for the great support and for the good times we have.

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“Nature makes the ultimate decision as to whether a flow will be laminar or

turbulent. There is a general principle in nature, that a system, when left to itself, will

always move toward its state of maximum disorder. To bring order to a system, we

generally have to exert some work on the system or expend energy in some manner.”

—John Anderson Jr.

(In Fundamentals of Aerodynamics)

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1. INTRODUCTION

Conventional non-renewable sources of energy, primarily fossil fuels, have

essentially been the key drivers of almost all modern economies. For over a century,

locomotion and hence the transportation industry has been one of the largest

consumer of oil, which has over time, led to a steady increase in demand for the same.

As new economies emerge and the existing well established ones continue to sustain

and diversify, as trade and business becomes more and more global, the transportation

sector will continue to play a major role in order to meet the demands of large scale

globalization. This translates into increased demand for oil which as a consequence of

reduced supply, leads to increase in fuel costs and thus higher operational costs.

The aviation industry in the last few decades has witnessed a huge surge,

attributed primarily to the growth of new and emerging economies in Asia. As cited in

a forecast report by the Boeing Company, passenger air traffic rose by almost 8

percent in 2010 after a slight slump in 2009. The world air cargo traffic saw an

estimated 24 percent growth in 2010. The global gross domestic product (GDP) is

projected to grow at an average of 3.3 percent per year for the next 20 years.

Reflecting this economic growth, worldwide passenger traffic will average 5.1 percent

growth and cargo traffic will average 5.6 percent growth over the forecast period. To

meet this increased demand for air transportation, the number of airplanes in the

worldwide fleet will grow at an annual rate of 3.6 percent, nearly doubling from

around 19,400 airplanes today to more than 39,500 airplanes in 2030 [1]. This surge

in global air traffic coupled with more airplanes going into operation is bound to

increase the demand for oil and hence increased costs.

The other issue associated with increased consumption of oil, an issue which

has over time become more serious and a major cause of worldwide concern, is that of

global warming. With the global surface temperatures on the rise, due to increased

green-house gas emissions resulting from the combustion of fossil fuels, nature and

the environment has greatly been affected. This issue, if ignored, can lead to grave

consequences.

If civil aviation is to continue growing, or even maintain its present level, the

industry must address the challenges posed by global warming and increased oil

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prices. One solution is provided by unconventional aircraft configurations that

significantly improve fuel efficiency and reduce greenhouse gas emissions.

However, modern aircraft are already remarkably efficient. A person traveling

from Seattle to New York on a Boeing 737-900 aircraft uses less fuel than one person

traveling the same distance in a gas-electric hybrid car. Indeed, the modern aircraft is

so highly optimized, that the desired performance improvements are likely not

possible using the conventional configuration. This has motivated the investigation of

unconventional aircraft and operations [2].

One of the many proposed configuration changes is the concept of active flow

control technology. The idea of active control is to expend a small amount of energy

to achieve a large gain in performance, such that the total amount of energy used is

lower.

1.1 Motivation

Biomimicry or biomimetics is the examination of nature, its models, systems,

processes, and elements to emulate or take inspiration from in order to solve human

problems. And this has given rise to biologically inspired engineering, a new

scientific discipline that applies biological principles to develop new engineering

solutions.

Right from the time man first tried to attain flight, all the way to the Airbus

A380; inspiration has always been drawn from the flight of birds, as they have been

considered to be a benchmark for aerodynamically efficient designs. The wing design

of the A380 for instance, was inspired by the way the eagle flies. The eagle’s wings

perfectly balance maximum lift with minimum length. It can manipulate the feathers

at its wingtips, curling them to create a ‘winglet,’ a natural adaptation that aids highly

efficient flight. The wingspan of the A380 had to be less than 80m in order to be able

to operate on existing international airports. This constraint gave rise to a very

important and a challenging question −How could it generate enough lift and still fit

inside airports? On a conventional wing, vortices created by high pressure air leaks

underneath mean the tips do not provide any lift. The wing had to be longer. To tackle

this problem, engineers designed small structural devices known as winglets, located

on the wing tips, which mimic the eagle’s feathers. By doing so, they were able to

limit the A380’s wingspan to just 79.8 m while generating the desired lift[3].

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A swift (Apus apus L.) adapts the shape of its wings to the immediate task at

hand− folding them back to chase insects, or stretching them out to sleep in

flight. During flight, they continually change the shape of their wings from spread

wide to swept back. When they fly slowly and straight on, extended wings carry

swifts 1.5 times farther and keep them airborne twice as long. Some ten Dutch and

Swedish scientists, based in Wageningen, Groningen, Delft, Leiden, and Lund, have

shown how wing morphing makes swifts such versatile flyers. Their study proves that

swifts can improve flight performance by up to three-fold, numbers that make the

wing morphing concept a potential candidate for next generation aircraft designs [4].

1.2 Historical perspective

Even before the official beginning of controlled human flight in 1903, radical

shape changing aircraft appeared and then disappeared, contributing little to aviation.

Clement Ader, a French inventor and engineer remembered primarily for his

pioneering work in aviation, conducted flight experiments in France as early as 1873

and proposed a wing morphing design as early as 1890. He developed ideas for the

future of aviation and described them in a short monograph published in 1909. In one

of his descriptions of the general military airplane he states -

“Whatever category airplanes might belong to, they must satisfy the following general conditions: their wings must be articulated in all their parts and must be able to fold up completely. When advances in aircraft design and construction permit, the frames will fold and the membranes will be elastic in order to diminish or increase the bearing surfaces at the wish of the pilot…”

In the beginning of the twentieth century many variable geometry aircraft

concepts were developed such as Ivan Makhonine’s telescopic wing concept in the

1930s that could change its wingspan by almost 62%. His concept later on inspired

Georges Bruner and Charles Gourdou to design a small aircraft named the G-11 C-1.

Apparently this airplane was never built, but its design had a wing whose area could

range between 11.4 and 17.2 square meters with spans between 6.76 and 11.4 meters.

The IS-1 fighter, designed by Nikitin-Shevchenko in 1932, morphed out-of-plane

from a bi-plane to a monoplane that was to operate at high speed. The XB-70

supersonic bomber also used a form of three-dimensional wing morphing. This design

used outer wing panel rotation to control L/D at both low subsonic and supersonic

speeds.

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The Mission Adaptive Wing (MAW) project produced wing camber changing

concepts and a flight demonstrator, the AFTI/F-111. Variable camber has been shown

to improve performance of fighter aircraft at all flight conditions. The F-16 and F-18

aircraft use discrete leading edge and trailing edge flap deflections to control camber,

although imperfectly. The MAW supercritical airfoil camber was controlled to create

optimum camber over a range of airspeeds for low subsonic to supersonic. This

control was achieved by continuously deformable leading and trailing edge deflection

using internal mechanisms to bending the chordwise shape as required. This allows

the wing to be a continuous surface at all times, unbroken by discrete surfaces[5] .

Figure 1 AFTI-F-111 Mission Adaptive Wing (MAW) in Flight

Lewis et al. patented a variable camber wing command system for varying

wing camber to optimize the wing lift-drag ratio during operation of the aircraft.

The invention included a means for sensing various flight conditions and

parameters during aircraft operation. The calculating means then calculates a desired

position for camber control surface to optimize the wing lift-drag ratio [6].

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Figure 2 Optimum wing cambers at different flight conditions

Figure 3 Variable leading and trailing edge wing design

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The NASA Aircraft Morphing program is an attempt to couple research across

a wide range of disciplines to integrate smart technologies into high payoff aircraft

applications. The program bridges research in seven individual disciplines and

Figure 4 Lewis et al variable camber wing command system

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combines the effort into activities in three primary program thrusts. System studies

are used to assess the highest payoff program objectives, and specific research

activities are defined to address the technologies required for development of smart

aircraft systems. The stated goal of the program is the development of smart devices

using active component technologies to enable self-adaptive flight for a revolutionary

improvement in aircraft efficiency and affordability. Aircraft Morphing is an

inherently multidisciplinary program, and has been built around a core discipline

based structure to provide the fundamental technology base. The integrated

disciplines are – Structures; Flow Physics; Systems and Multidisciplinary

Optimization; Integration; Controls; Acoustics; and Materials [7] .

The Boeing Company has also been researching in the field of morphing

wings and has published a number of patents pertaining to this subject. Dockter et al.

have invented a morphing wing design that includes a rigid internal core, an

expandable spar surrounding it and a plurality of elastomeric bladders. An external

fiber mesh overlay covers the elastomeric bladders to provide a smooth wing surface.

The plurality of elastomeric bladders is expandable through the introduction of

Figure 5 Artist’s rendering of NASA’s Morphing Airplane

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increased air pressure such that the profile of the geometric morphing wing can be

modified [8].

In another patent, Sankrithi et al. have invented systems and methods for

providing variable geometry winglets to an aircraft. In one of the embodiments, a

method for adapting an aircraft to a plurality of flight conditions comprises providing

a winglet to each wing. The winglet includes a body portion which in turn includes at

least one of a deflectable control surface, a shape memory alloy (SMA) bending plate,

and a SMA torque tube. Once the aircraft is moving through the atmosphere, the

method further includes providing a winglet modification signal. The winglet is then

modified based on the modification signal to at least one of reduce induced drag and

redistribute a wing load [9].

The Decadal Survey of Civil Aeronautics conducted by the National Research

Council (NRC), has ranked the use of adaptive materials and morphing structures as

one of high-priority research and technology (R&T) challenges. This subject has

received equal priorities by NASA and at the National level. It has been featured

second in the list of R&T challenges in the area of Materials and Structures.

DARPA’s (Defense Advanced Research Projects Agency) Morphing Aircraft

Structures Program has sponsored several wind tunnel experiments at NASA Langley

Research Center’s Transonic Dynamics Tunnel. These produced a wide range of

experience and identified innovative aircraft and rotorcraft concepts and critical

materials technologies for future work[10] .

1.3 Problem Statement

There are essentially two factors limiting overall aircraft or aircraft-component

performance—the development of the boundary layer and the interaction of the

boundary layer with the outer flow field, exacerbated at high speeds by the occurrence

of shock waves, and the fact that flight and freestream conditions may change

considerably during an aircraft mission while the actual aircraft is only designed for

multiple but fixed design points rendering such an aircraft a compromise with regard

to performance. Flow control and adaptive wings, which are naturally

complementary, adjust the flow development to the prevailing flight condition and

have, therefore, the potential to greatly improve aircraft and aircraft-component

performance[11].

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Commercial airplanes generally follow specific flight profiles consisting of

take-off, climb, cruise, descend and landing. These flight profiles essentially change

the freestream conditions in which the aircrafts operate. Furthermore, over the course

of the flight, the required lift force changes as the fuel gets consumed. The

conventional fixed wing designs account for these requirements by catering to

multiple but fixed design points which, however, compromise the overall flight

performance.

The two dominant flight parameters influencing aerodynamic forces of a given

wing are—airspeed and altitude. It is desirable that the aerodynamic efficiency of the

wing be at its maximum (subject to certain constraints) throughout the flight

envelope. Employing adaptive wing technology where the effective wing geometry

adjusts to the changing flow and load requirements, allows to fully exploring the

aerodynamic flow potential at each point of the flight envelope, thus resulting in

aerodynamic performance gains during cruise and maneuver, and, furthermore, most

likely improving structural designs.

A solution to this problem is the morphing wing concept in which the wing

shape is altered depending on the existing flight condition. The primary motive for

altering wing geometry is to improve airfoil efficiency in off-design flight regimes. A

particular, more constrained implementation of this concept can be seen in most

modern aircraft designs existing today and it takes the form of flaps which change the

wing area and/or effective camber. A polymorph wing (variable planform) and

variable pitch or incidence are also proven methods of wing adaptation and can be

found in some multi-role combat aircrafts [12]. However these forms essentially allow

only rigid or constrained deformations in that, the deformations are discrete or

discontinuous such as the case of trailing edge flaps or leading edge slats deployed

while take-off and landing in transport aircrafts. In the remaining portion of the flight

the wing section is predominantly constant whereas the free-stream conditions

continue to change with change in airspeed and altitude thus compromising all round

performance. In order to improve the performance over a greater portion of the flight

envelope, it is desirable that the wing shape undergoes considerable deformation

thereby morph into a shape that achieves desirable low-drag pressure distributions.

The problem definition for this thesis is therefore to develop an optimization

framework that could be used to develop single-point optimized airfoil shapes at

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discrete locations within the flight envelope. These shapes could then be embedded in

a control system architecture that could signal actuators to deform the existing airfoil

shape into the desired one that has been optimized for the corresponding freestream

conditions of the aircraft.

1.4 Review of Literature

A plethora of research and studies has been performed in the area of

aerodynamic shape optimization for the design of optimal airfoil shapes. This research

area has undoubtedly been dominated by the pioneering works of Antony Jameson

who has extensively used control theory for airfoil and wing design [13-16]. Here the

shape optimization problem is treated as a control problem, in which the airfoil

(and/or wing) is treated as a device which controls the flow to produce lift with

minimum drag, while meeting other requirements such as low structure weight,

sufficient fuel volume, and stability and control constraints. The theory of optimal

control of systems is applied which is governed by partial differential equations with

boundary control, in this case through changing the shape of the boundary. Here the

Frechet derivative of the cost function is determined via the solution of an adjoint

partial differential equation, and the boundary shape is then modified in a direction of

descent. This process is repeated until an optimum solution is approached.

E. Stanewsky [11] provides a thorough review of the adaptive wing

technology. Here, various concepts on complete wing adaptation both for cruise-drag

minimization as well as for maneuver load control for combat aircrafts are discussed.

The concepts that have been discussed are the adaptive variable camber trailing edges,

leading edges and transonic contour bumps, and also combinations of the three. He

also discusses optimum number of linear actuators based on the drag reduction

achieved and the percentage flexing of the surfaces.

Fred Austin et al. have developed optimum wing cross-sectional profiles for a

hypothetical fixed-wing version of the F-14 aircraft at two transonic cruise conditions

[17]. Jameson’s control-theory based optimum aerodynamic design approach is used

for the analysis at critical operating conditions where shape modifications might

significantly improve the aerodynamic performance. Based on the analysis it is shown

that only small, potentially achievable, adaptive modifications to the profile are

required. A general method is developed to adaptively deform the structures to

25

desired shapes. The method is investigated by a finite element analysis of an adaptive

rib and by experimental investigations on a rib model. Open loop experiments of the

unloaded structure show that commanded shapes can be achieved.

David Zingg et al. have used a Newton-Krylov based aerodynamic

optimization algorithm to assess an adaptive airfoil concept for drag reduction at

transonic speeds [18]. The lift-constrained drag minimization problem is used as the

objective function. The airfoil geometry is parameterized using B-splines and the

compressible Navier-Stokes equations are solved with a Newton-Krylov method. In

this study, a baseline is designed to produce low drag at a fixed lift coefficient over a

range of Mach numbers. This airfoil is then compared with a sequence of nine airfoils,

each designed to be optimal at a single operating point in the Mach number range. It

is found that shape changes of less than 2% chord lead to drag reduction of 4 to 6%

over a range of Mach numbers from 0.68 to 0.76. If the shape changes are restricted to

the upper surface only, then changes of less than 1% chord lead to a drag reduction of

3 to 5%.

Namgoong in his research has developed appropriate problem formulations

and optimization strategies to design an airfoil for morphing aircraft that include the

energy required for shape change [19]. Here the relative strain energy needed to

change from one airfoil shape to another is included as an additional design objective

along with a drag design objective, while constraints are enforced on the lift. Solving

the resulting multi-objective problem generates a range of morphing airfoil designs

that represent the best tradeoffs between aerodynamic performance and morphing

energy requirements. The airfoil parameterization is done here using Hicks-Henne

shape functions. To formulate the problem as a multi-objective problem, a strain

energy model is developed that depends upon the concept of linearly elastic springs.

The Genetic Algorithm (GA) is used for solving the multi-objective optimization

problem.

Gamboa et al. in their work have designed a morphing wing concept for a

small experimental unmanned aerial vehicle to improve the vehicle’s performance

over its intended speed range [20]. The wing is designed with a multidisciplinary

design optimization tool, in which an aerodynamic shape optimization code coupled

with a structural morphing model is used to obtain a set of optimal wing shapes for

minimum drag at different flight speeds. The optimization studies reveal that wing

26

drag reduction of up to 30% can be achieved by morphing the wing, thus representing

the important performance improvements in off-design conditions.

Ursache in his research has designed adaptive structures for achieving global

multi-shape morphing aerodynamic configurations, by using slender structures. A

heuristic approach is proposed that enables morphing through a range of stable

cambered airfoils to achieve the aerodynamic properties for different maneuvers, with

the benefit of low powered actuation control [21]. Ünlüsoy in his work has performed

structural design and analysis of a wing having mission adaptive control surfaces. The

wing is designed to withstand a maximum aerodynamic loading of 5g during

maneuver. The structural model of the wing is developed using MSC/PATRAN and

the analysis is carried out using MSC/NASTRAN [22].

1.5 Thesis Objectives

As outlined in the previous section, the goal is to achieve a set of airfoil shapes

that have been optimized for specific points in the flight envelope, viz at specific

airspeeds and altitudes.

This, as is evident from the aforementioned description, essentially translates

into an optimization problem. Thus, aerodynamic shape optimization forms the core

and one of the most important parts of this thesis which, in the subsequent chapters,

will be addressed in greater detail.

The shape optimization process is to be carried out for different operating

points within the flight envelope. Only those operating points are chosen which have a

greater probability of being visited by a typical subsonic transport airplane. A

comparative analysis is performed in which the aerodynamic performance of a

Boeing-737 classic airfoil at various operating points is compared to that of a single

point optimized airfoil at the same operating points.

Chapter 2 provides an overview of the theories required for the shape

optimization process. This includes the formulation of the optimization problem,

shape parameterization to obtain the design variables, various objective functions that

are employed for shape optimization problems and the constraints on the optimization

process. The governing equations that form an important part of any fluid dynamics

27

problem alongwith the procedure for analysis using Computation Fluid Dynamics

(CFD) solvers are also discussed.

Chapter 3 introduces the Iterative Response Surface Optimization

methodology for carrying out the shape optimization process. Here the various steps

involved such as the Design of Experiments (DoE), response surface construction and

the iterative improvement of this model are discussed. Also included is the model

testing methods and the mathematical optimization of the response model. This is

followed by the steps for constructing the CFD grid using GridPro® and the flow

analysis using ANSYS-Fluent® along with its validation are discussed towards the

end of the chapter.

Chapter 4 primarily discusses the results of this study. Here the lift-moment

constrained drag minimization scheme employed in the optimization process is

discussed followed by the case study. Also discussed are the operational flight

envelopes of typical transonic aircrafts. The RAE 2822 design study is carried out to

validate the optimization algorithm developed. The adaptive airfoil concept is

demonstrated using a Boeing-737 classic airfoil at three steady flight operating

conditions. The aerodynamic performance of the adaptive airfoil is then compared to

that of the baseline airfoil.

Chapter 5 summarizes and concludes the research conducted in this thesis

along with shedding some light on possible improvements and more advanced study

that can be carried out on this subject in the future.

28

2. OPTIMIZATION AND CFD

2.1 Introduction to Optimization

Nature optimizes. Physical systems tend to a state of minimum energy. The

molecules in an isolated chemical system react with each other until the total potential

energy of their electrons is minimized. Rays of light follow paths that minimize their

travel time.

On similar lines people optimize. Manufacturers aim for maximum efficiency

in the design and operation of their production processes. Engineers adjust parameters

to optimize the performance of their designs [23].

Optimization is the act of obtaining the best result under given circumstances.

In design, construction, and maintenance of any engineering system, engineers have

to take many technological and managerial decisions at several stages. The ultimate

goal of all such decisions is either to minimize the effort required or to maximize the

desired benefit [24]. Optimization can be considered as an important tool in decision

science and in the analysis of physical systems. To make use of this tool, an objective

must be defined. The objective is a quantitative measure of the performance of the

system under study. This could be profit, time, potential energy, or any quantity or

combination of quantities that can be represented by a single number. The objective

depends on certain characteristics of the system, called variables or unknowns. The

goal is to find values of the variables that optimize the objective. Often the variables

are restricted, or constrained, in some way.

The process of identifying objective, variables, and constraints for a given

problem is known as modeling. Construction of an appropriate model is the first and

one of the most important step in the optimization process. Once the model has been

formulated, an optimization algorithm can be used to find its solution, usually with

the help of a computer. There is no universal optimization algorithm but rather a

collection of algorithms, each of which is tailored to a particular type of optimization

problem. The responsibility of choosing the algorithm that is appropriate for a specific

application often falls on the user. This choice is an important one, as it may

determine whether the problem is solved rapidly or slowly and, indeed, whether the

solution is found at all.

29

Mathematical Formulation 2.1.1

Mathematically, optimization is the minimization or maximization of a

function subject to constraints on its variables. The optimization problem can be

written as follows:

min𝑥∈𝑅𝑛

𝑓(𝐗) subject to 𝑔𝑖(𝐗) = 0, 𝑖 = 1, 2, …𝑚

ℎ𝑘(𝐗) ≤ 0, 𝑘 = 1, 2, … 𝑝

- X is the vector of variables, also called unknowns or parameters;

- f is the objective function, a scalar function of X that has to be optimized.

- gi and ℎ𝑘 are the constraint functions, which are scalar functions of X that

define certain equalities and inequalities that X must satisfy.

- 𝑚 and p are the number of equalities and inequalites constraints.

Characteristics of Optimization Algorithms 2.1.2

Optimization algorithms are iterative. They begin with an initial guess of the

variable X and generate a sequence of improved estimates, called iterates, until they

terminate, hopefully at a solution. The strategy used to move from one iterate to the

next distinguishes one algorithm from another. Most strategies make use of the values

of the objective function f, the constraint functions 𝑔𝑖 and/or ℎ𝑘, and possibly the first

and second derivatives of these functions. Some algorithms accumulate information

gathered at previous iterations, while others use only local information obtained at the

current point.

Regardless of these specifics, good algorithms should possess the following

properties:

- Robustness: They should perform well on a wide variety of problems in

their class, for all reasonable values of the starting point.

- Efficiency: They should not require excessive computer time or storage.

- Accuracy: They should be able to identify a solution with precision,

without being overly sensitive to errors in the data or to the arithmetic

rounding errors that occur when the algorithm is implemented on a

computer.

These goals may conflict. For example, a rapidly convergent method for a

large unconstrained nonlinear problem may require too much computer storage. On

30

the other hand, a robust method may also be the slowest. Tradeoffs between

convergence rate and storage requirements, and between robustness and speed, and so

on, are central issues in numerical optimization.

2.2 Sequential Quadratic Programming

Sequential quadratic programming (SQP) is an iterative method for nonlinear

optimization. SQP methods are used on problems for which the objective function and

the constraints are twice continuously differentiable.

The sequential quadratic programming is one of the most recently developed

and perhaps one of the best methods of optimization. The method has a theoretical

basis that is related to (1) the solution of a set of nonlinear equations using Newton’s

method, and (2) the derivation of simultaneous nonlinear equations using Kuhn–

Tucker conditions to the Lagrangian of the constrained optimization problem.

The SQP Technique 2.2.1

Consider a nonlinear optimization problem as below:

min𝐗𝑓(𝐗)

Subject to ℎ𝑘(𝐗) ≤ 0, 𝑘 = 1, 2, … ,𝑝

The Lagrange formulation of the above problem then becomes

𝐿(𝐗, 𝜆) = 𝑓(𝐗) + 𝜆𝑘

𝑝

𝑘=1

ℎ𝑘(𝐗)

Here 𝜆𝑘 are the Lagrange multipliers for each of the kth constraints. It is

assumed that the bound constraints have been expressed as inequality constraints.

Now in the SQP technique, the above optimization problem can be solved iteratively

by solving a Quadratic Programming (QP) problem which is stated as—

min∆𝐗

𝑄 = ∇𝑓𝑇∆𝐗 +12∆𝐗𝑇[∇2𝐿]∆𝐗

subject to

𝑔𝑗 + ∇𝑔𝑗𝑇∆𝐗 ≤ 0, j = 1,2, … , m

ℎ𝑘 + ∇ℎ𝑘𝑇∆𝐗 = 0,𝑘 = 1, 2, … ,𝑝

31

The Langrange formulation is given by

𝐿 = 𝑓(𝐗) + 𝜆𝑗

𝑚

𝑗=1

𝑔𝑗(𝐗) + 𝜆𝑚+𝑘

𝑝

𝑘=1

ℎ𝑘(𝐗)

Since the minimum of the augmented Lagrange function is involved, the SQP

method is also known as the projected Lagrangian method.

Solution Procedure 2.2.2

As in the case of Newton’s method of unconstrained minimization, the

solution vector ∆𝐗 is treated as the search direction, S, and the quadratic programming

sub-problem (in terms of the design vector S) is restated as:

min𝐒𝑄(𝐒) = ∇𝑓(𝐗)𝑇𝐒 +

12𝐒𝑇[𝐻]𝐒

Subject to

𝑔𝑗(𝐗) + ∇𝑔𝑗(𝐗)𝑇𝐒 ≤ 0, j = 1,2, … , m

ℎ𝑘(𝐗) + ∇ℎ𝑘(𝐗)𝑇𝐒 = 0, 𝑘 = 1, 2, … ,𝑝

where [H] is a positive definite matrix that is taken initially as the identity

matrix and is updated in subsequent iterations so as to converge to the Hessian matrix

of the Lagrange function 𝐿. The above sub-problem is q quadratic programming

problem and can be solved using any optimization technique. Once the search

direction S, is found the design variables can be updated as

𝑿𝑗+1 = 𝑿𝑗 + 𝛼 ∙ 𝐒

Here 𝛼 is the optimal step length along the direction S and can be determined

by an appropriate line search procedure so that a sufficient decrease in a merit

function is obtained. The matrix [𝐻] is a positive definite approximation of the

Hessian matrix of the Lagrangian function.

Updating the Hessian Matrix 2.2.3

At each major iteration a positive definite quasi-Newton approximation of the

Hessian of the Lagrangian function, [H], is calculated using the BFGS (Broyden-

Fletcher-Goldfarb-Shanno) method, where 𝜆𝑖, 𝑖 = 1, … ,𝑚, is an estimate of the

Lagrange multipliers.

32

𝐻𝑘+1 = 𝐻𝑘 +𝑞𝑘𝑞𝑘𝑇

𝑞𝑘𝑇𝑠𝑘−𝐻𝑘𝑇𝑠𝑘𝑇𝑠𝑘𝐻𝑘𝑠𝑘𝑇𝐻𝑘𝑠𝑘

,

Where

𝑠𝑘 = 𝐗k+1 − 𝐗𝑘

𝑞𝑘 = ∇𝑓(𝐗𝑘+1) + 𝜆𝑖

𝑚

𝑖=1

𝑔𝑖(𝑿𝑘+1)− ∇𝑓(𝑿𝑘) + 𝜆𝑖

𝑚

𝑖=1

𝑔𝑖(𝑿𝑘)

2.3 Design Variables

As mentioned in the previous section, design variables are quantities or

quantifiable characteristics on which the performance of the system depends. These

are quantities which almost completely define the system being analyzed.

From an aerodynamic shape optimization point of view, the vector of design

variables, X, primarily contains parameters that control the shape of the airfoil. These

are geometrical entities that are relatively few in number and fully parameterize the

airfoil shape. The angle of attack, which is the angle that the airfoil chordline makes

with the direction of airflow, may be used as an additional design variable.

Here, B-Spline control points are used to define the airfoil shape. These

control points along with angle of attack form the vector of design variables for the

purpose of optimization. The following sub-sections describe the process of shape

parameterization using B-Spline curves.

B-Spline Curves Introduction 2.3.1

NURBS, or Non-Uniform Rational B-Splines, are the standard for describing

and modeling curves and surfaces in computer aided design and computer graphics.

They are used to model everything from automobile bodies and ship hulls to animated

characters.

B-Splines belong to the class of parametric curves. Parametric curve

representations of the form: 𝑥 = 𝑓(𝑡); 𝑦 = 𝑔(𝑡); 𝑧 = 𝑔(𝑡)

where t is the parameter, are extremely flexible. They are axis independent,

easily represent multiple-valued functions and infinite derivatives, and have additional

degrees of freedom compared to either explicit or implicit formulations.

33

B-Spline Formulation 2.3.2

The general mathematical formulation of a B-spline curve given the order,

control point coordinates and a parameter set, is as follows:

𝐶𝑥𝑡𝑗 = ∑ 𝑁𝑖,𝑘𝑡𝑗 ∙ 𝑃𝑖𝑥𝑛+1𝑖=1

𝐶𝑦𝑡𝑗 = ∑ 𝑁𝑖,𝑘𝑡𝑗 ∙ 𝑃𝑖𝑦𝑛+1

𝑖=1

The notations used in the above formulation are as described below:

- (Cx, Cy) are the Cartesian coordinates of the B-Spline curve.

- t is the curve parameter,

- (𝑃𝑖𝑥 ,𝑃𝑖𝑦) are the co-ordinates of the B-Spline control points,

- (n+1) is the number of control points,

- k is the order of the curve (degree is k −1),

- j is the index for the curve parameter u,

- 𝑁𝑖,𝑘 are the B-Spline Basis functions defined by the Cox-de Boor

recurrence relations as below:

𝑁𝑖,1𝑡𝑗 = 1,0,

if 𝑈𝑖 ≤ 𝑡𝑗 < 𝑈𝑖+1 otherwise

𝑁𝑖,𝑘𝑡𝑗 = 𝑡𝑗 − 𝑈𝑖

𝑈𝑖+𝑘−1 − 𝑈𝑖𝑁𝑖,𝑘−1𝑡𝑗 +

𝑈𝑖+𝑘 − 𝑡𝑗𝑈𝑖+𝑘 − 𝑈𝑖+1

𝑁𝑖+1,𝑘−1𝑡𝑗 ]

- The vector 𝑈𝑖 represents a normalized uniform knot sequence given by:

𝑈𝑖 = 0,

(𝑈𝑖−1 + 1) (𝑈(𝑛+1)+𝑘⁄ ,𝑈𝑖−1 (𝑈(𝑛+1)+𝑘⁄ ,

for 1 < 𝑖 ≤ 𝑘 for 𝑘 < 𝑖 < (𝑛 + 1) + 2

for (𝑛 + 1) + 2 ≤ 𝑖 ≤ (𝑛 + 1) + 𝑘

Airfoil parameterization using B-Spline curves 2.3.3

The distance along the B-spline curve is represented by the parameter value tj

for each point j on the surface of the airfoil. The initial values of tj are given by

𝑡1 = 0

𝑡𝑗 =∑ (𝑥𝑑𝑎𝑡𝑎𝑖−𝑥𝑑𝑎𝑡𝑎𝑖−1)2+(𝑦𝑑𝑎𝑡𝑎𝑖−𝑦𝑑𝑎𝑡𝑎𝑖−1)2𝑗𝑖=2

𝐿𝑇… . 𝑗 ∈ [2,𝑁𝑑𝑣𝑡𝑣]

34

𝐿𝑇 = 𝑥𝑑𝑣𝑡𝑣𝑖 − 𝑥𝑑𝑣𝑡𝑣𝑖−12

+ 𝑦𝑑𝑣𝑡𝑣𝑖 − 𝑦𝑑𝑣𝑡𝑣𝑖−12

𝑁𝑑𝑎𝑡𝑎

𝑖=2

… . (𝑥𝑑𝑣𝑡𝑣𝑖 ,𝑦𝑑𝑣𝑡𝑣𝑖)

∈ [0,1]

Here:

- Ndata, is the number of data points of the airfoil to be parameterized,

- (xdata,ydata) are the airfoil coordinates.

- LT is the sum of the chord distances for all the data points.

At the onset of the optimization procedure, it is necessary to determine the

location of the B-spline control points that best approximate the initial airfoil shape.

In order to locate the control point co-ordinates a curve-fitting strategy must be

employed. The curve fitting approach employed here implements a curve parameter

correction algorithm [25]. The algorithm consists of the following two separate parts

- Initial parameterization

- Fitting/reparametrization loop

Initial parameterization yields an initial set of control-point coordinates that

roughly approximates the airfoil shape. If a data point lies on the B-spline curve, then

it must satisfy—

𝐶𝑥(𝑡1) = 𝑁1,𝑘(𝑡1) ∙ 𝑃1𝑥 + 𝑁2,𝑘(𝑡1) ∙ 𝑃2𝑥 + ⋯+ 𝑁𝑛+1,𝑘(𝑡1) ∙ 𝑃𝑛+1𝑥 𝐶𝑦(𝑡1)

= 𝑁1,𝑘(𝑡1) ∙ 𝑃1𝑦 + 𝑁2,𝑘(𝑡1) ∙ 𝑃2

𝑦 + ⋯+ 𝑁𝑛+1,𝑘(𝑡1) ∙ 𝑃𝑛+1𝑦

𝐶𝑥(𝑡2) = 𝑁1,𝑘(𝑡2) ∙ 𝑃1𝑥 + 𝑁2,𝑘(𝑡2) ∙ 𝑃2𝑥 + ⋯+ 𝑁𝑛+1,𝑘(𝑡2) ∙ 𝑃𝑛+1𝑥 𝐶𝑦(𝑡2)

= 𝑁1,𝑘(𝑡2) ∙ 𝑃1𝑦 + 𝑁2,𝑘(𝑡2) ∙ 𝑃2

𝑦 + ⋯+ 𝑁𝑛+1,𝑘(𝑡2) ∙ 𝑃𝑛+1𝑦 ⋮

𝐶𝑥𝑡𝑗 = 𝑁1,𝑘𝑡𝑗 ∙ 𝑃1𝑥 + 𝑁2,𝑘𝑡𝑗 ∙ 𝑃2𝑥 + ⋯+ 𝑁𝑛+1,𝑘𝑡𝑗 ∙ 𝑃𝑛+1𝑥 𝐶𝑦𝑡𝑗

= 𝑁1,𝑘𝑡𝑗 ∙ 𝑃1𝑦 + 𝑁2,𝑘𝑡𝑗 ∙ 𝑃2

𝑦 + ⋯+ 𝑁𝑛+1,𝑘𝑡𝑗 ∙ 𝑃𝑛+1𝑦

where 2 ≤ 𝑘 ≤ 𝑛 + 1 ≤ 𝑗. This system of equations is more compactly written in

matrix form as

[𝐶𝑥] = [𝑁] ∙ [𝑃𝑥]

𝐶𝑦 = [𝑁] ∙ [𝑃𝑦]

Recalling that a matrix times its transpose is always square, the control

polygon for a B-spline curve that approximates the data is given by

35

[𝑁]𝑇[𝐶𝑥] = [𝑁]𝑇[𝑁] ∙ [𝑃𝑥]

∴ [𝑃𝑥] = [𝑁]𝑇[𝑁]−1∙ [𝐶𝑥]

Likewise [𝑁]𝑇𝐶𝑦 = [𝑁]𝑇[𝑁] ∙ 𝑃𝑦

∴ 𝑃𝑦 = [𝑁]𝑇[𝑁]−1∙ 𝐶𝑦

Provided that the order of the B-spline basis k, the number of control polygon

vertices n + 1, and the parameter values along the curve are known, then the basis

functions 𝑁𝑖,𝑘𝑡𝑗 and hence the matrix [N] can be obtained. Within the restrictions

2 ≤ 𝑘 ≤ 𝑛 + 1 ≤ 𝑗 , the order and number of polygon vertices are arbitrary.

In the reparameterization stage, the parameter values are improved by

replacing the previous parameters by new parameter values which are obtained after

implementing parameter correction formula. The algorithm is explained as follows.

The basic aim here is to fit a B-Spline curve that will approximate N measured

data points in a Least Squares Method sense. This leads to minimization problem in

which the objective is to find an optimal set of parameter values 𝑡𝑖 producing an

optimal approximating spline C(𝑡𝑖) with minimal distances to the data points Di. The

objective function is—

𝐸𝑖 = ‖𝐷𝑖 − 𝐶(𝑡𝑖)‖2 → 𝑚𝑖𝑛𝑁

𝑖=1

Where 𝐶(𝑡𝑖)is the B_spline curve point at 𝑡𝑖 and 𝐷𝑖 is the corresponding

measured data point. In curve parameterization strategies, the distance vectors are

generally not perpendicular to the surface. This means an arbitrary error due to the

parameterization has been minimized. This parameterization scheme can be improved

by changing the aim to approximate the shortest distances, which means sequence of

new parameter values ti will be constructed with the goal that the corresponding error

vectors Ei converge in general to the normal of the approximation curve.

C(ti) ∆𝑡𝑖

Ei

Di

Figure 6 Reparameterization

36

Therefore, the curve at each point C(ti) is replaced by the tangent as described

in Fig.2.1, by projecting the error vector Ei on the tangent and obtain ∆𝑡𝑖 as a measure

for changing the parameter values 𝑡𝑖 in direction of the parameter values of the

perpendicular from Di to C(𝑡𝑖). One of the methods for minimizing the local error

vector is as suggested by Hoschek et al [26]. The idea is to minimize 𝐸𝑖2 =

𝐷𝑖 − 𝐶(𝑡𝑖)2 . This after differentiation leads to: 𝑓 ≔ 𝐷𝑖(𝑡𝑖) = 0

Now if the Newton iteration formula is used to compute a zero of f, the

correction formula is obtained as

∆𝑢𝑖 = −𝑓𝑓

=−𝐷𝑖(𝑡𝑖)

𝐷𝑖(𝑡𝑖) − ⟨𝐷𝑖|𝐷𝑖⟩= −

(𝐶(𝑡𝑖) − 𝐷𝑖) ∙ (𝑡𝑖)(𝐶(𝑡𝑖) − 𝐷𝑖) ∙ (𝑡𝑖) + (𝑡𝑖)2

Therefore the improved parameter becomes 𝑡𝑖′ = 𝑡𝑖 + ∆𝑡𝑖

This reparameterization step is repeated till the desired accuracy is achieved.

The least-squares problem is re-evaluated with the final parameter vector in order to

obtain a better set of control points. This procedure typically converges within a few

hundred iterations and do not require significant computational effort.

In addition, the airfoil geometry to be parameterized is split into two —upper

surface and lower surface. This way the parameterization is split into two parts—one

for each of the split surfaces. So a set of B-Spline control points and a set of curve

parameters control one of the two surfaces of the airfoil, and a different pair of sets

controls the other airfoil surface. The shape control is hence independent for the upper

and lower surfaces, thus offering greater flexibility.

The parameterization is carried out in a way so that the approximating B-

Spline curve passes through the first and the last control points, which form the

trailing edge and leading edge of the airfoil respectively. This ensures that B-Spline

curve always passes through the leading and trailing edges thus maintaining a fixed

chord length. Also the B-spline curve maintains tangency to the control points

polygon at the first and last control points. This behavior of the B-spline curve is as a

result of the formulation of the uniform knot vector which has been described in

Section 2.2.2.

Each of the two airfoil surfaces is controlled individually by seven B-spline

control points and a set of curve parameters. The numbering of the control points is

37

done counter-clockwise beginning with the upper surface and then moving on the

lower surface. For the upper surface the numbering begins with control point number

1 at the trailing edge ending with the control point number 7 at the leading edge. For

the lower surface the counting begins with control point number 8 at the leading edge

ending with control point number 14 at the trailing edge. The first and the seventh

control points remain fixed for the upper surface as they represent the trailing and

leading edges respectively, as described above. Likewise, eighth and the fourteenth

control point remain fixed for the lower surface. Control points 1 and 14 (trailing

edge); 7 and 8 (leading edge) are coincident. For the upper surface, control point 6

and 7 share the same x-ordinate. Likewise, for the lower surface, control point 8 and 9

share the same x-ordinate. This is done so as to ensure a smooth and continuous

transition from the upper surface to the lower surface at the leading edge. By doing

so, there are no cusps or kinks formed, as both, the upper and lower surfaces have the

same tangency at the leading edge control points. It may be noted here that, as the

parameterization is carried out individually for the upper and lower surface of the

airfoil, there are two different sets of curve parameters ui (one for each surface). These

two sets remain fixed throughout the optimization process.

As an example, the parameterization of RAE 2822 airfoil is shown in the

above figure. The original airfoil data is split into upper and lower surfaces. The

parameterization is then carried out using 14 control points (7 for each surface) and

curve order of 5. CPs 1, 14, 7 and 8 are fixed. CPs 6 and 9 share the same x-ordinates

Figure 7 B-Spline parameterization of RAE 2822 airfoil

38

as those of CPs 7 and 8. The algorithm fits well to the airfoil data as is evident from

the above figure.

Design Variables Vector 2.3.4

The design variables vector is formed by concatenating the y-ordinates of all

the free controls points— CPs 2 through 6 (for the upper surface) and CPs 9 through

13 (for the lower surface). In addition to the ten control points, the angle of attack is

included as the eleventh design variable. As is evident, the CPs are allowed to move

only along the y-direction and so the x-ordinates remain fixed throughout the

optimization process. However, if required, the x-ordinates can readily be added to

the design variables vector.

2.4 Objective Functions

Objective function as is described earlier is a scalar valued function that

determines the performance of the system. From an aerodynamic shape optimization

point of view, the objective is primarily to minimize the drag or maximize the

aerodynamic efficiency.

There are various objective functions that cater to the problem of aerodynamic

shape optimization as discussed below:

Inverse design 2.4.1

In the inverse design formulation, the objective is to reach a shape such that

the pressure distribution over its surface matches a target pressure distribution.

Mathematically, the objective is represented as

𝐽 = 12 𝐶𝑃𝑠𝑢,𝑙 − 𝐶𝑃∗𝑠𝑢,𝑙

2𝐿.𝐷

𝑇.𝐷𝑑𝑠𝑢,𝑙

where 𝐶𝑃∗ represents the target pressure coefficient as function of 𝑠𝑢,𝑙 is the

distance along the upper and lower surfaces measured from the Teading edge (L.E) to

the Leading Edge. By minimizing 𝐽, the optimizer finds the shape of the airfoil that, in

the least-squares sense, best matches the target pressure distribution. A drawback

associated with this type of formulation is that the target pressure distribution needs to

be provided by the user. This requires considerable amount of experience and

knowledge of the flow physics on the part of the user.

39

Maximizing Aerodynamic Efficiency 2.4.2

This is a relatively easier function, where the objective is to simply maximize

the aerodynamic efficiency which is the ratio of the lift to the drag. The problem is

generally formulated as the minimization of the inverse of the aerodynamic

efficiency.

𝐽 = 𝐶𝐷𝐶𝐿

This objective is suitable for problems which do not require a constraint on the

lift force or moments.

Lift-Constrained Drag Minimization 2.4.3

In the lift-constrained drag-minimization formulation, the objective is to

reduce drag while maintaining the desired lift force. This type of formulation is

suitable for problems involving shape optimizations for airplanes where it is

advantageous to reduce the drag while ensuring that the lift generated is greater than

and approximately equal to the weight of the airplane in steady flight.

Mathematically, the problem is formulated as

𝐽 =

⎩⎪⎨

⎪⎧𝜔𝐿 1 −

𝐶𝐿𝐶𝐿∗2

+ 𝜔𝐷 1 −𝐶𝐷𝐶𝐷∗2

if 𝐶𝐷 > 𝐶𝐷∗

𝜔𝐿 1 −𝐶𝐿𝐶𝐿∗2

otherwise

where 𝐶𝐿∗ and 𝐶𝐷∗ represent the target lift and drag coefficients, respectively.

The coefficients 𝜔𝐿 and 𝜔𝐷 are the weights on the lift and drag coefficients

respectively. 𝐶𝐷∗ is often selected low enough to be considered unattainable. This

objective function consists of two competing objectives, namely attaining the target

lift and minimizing drag. Achieving both simultaneously is difficult, since a decrease

in drag typically corresponds to a decrease in lift. As the lift is a constraint on the

design, it is important to ensure that the desired coefficient of lift is reached. To do

this, either the weight on the lift coefficient objective must be very high (which will

reduce the amount by which drag is decreased), or the target lift entered into the

optimization program must be a carefully selected amount greater than the actual

target lift [27].

40

To determine how much greater the target lift coefficient given to the program

should be than the actual desired lift coefficient requires some experience. Some level

of experience is also required to successfully select a value for 𝐶𝐷∗ , and selecting

appropriate parameters can take several trials.

One of the advantages of using this kind of formulation is that additional terms

can be readily added to the problem. For example the moment coefficient along with a

suitable weight, can be added in the same manner as the lift coefficient. Also this

formulation falls into the category of unconstrained minimization in which the

constraint is included within objective function as a penalty term. Here the constraint

on the lift coefficient is added as penalty term in the objective function. Doing so

avoids an extra formulation of a constraints equation thereby reducing the burden on

the optimization process.

In this thesis, the lift- moment constrained drag-minimization objective

function will be employed.

2.5 Flow Equations

In order to obtain values of lift, drag and moment coefficients as well as the

surface pressure distribution, which are needed to evaluate the objective functions, it

is necessary to compute the flow or in other words determine the solution of the

airflow over the surfaces.

From fluid mechanics, there are fundamental equations that govern the

behavior of any fluid that flows over a submerged surface. These are the governing

equations that derived using the three fundamental physical principles that central to

the macroscopic observations of nature, which are:

- Conservation of mass, i.e., mass can neither be created nor be destroyed.

- Newton’s Second Law, i.e., force is directly proportional to the rate of

change of momentum, and

- Conservation of energy, i.e., energy can only change from one form to

another.

These principles have led to the formulation of complex differential equations

that essentially describe the governing of fluid motion. These equations are:

41

- Continuity and Momentum Equations which are known as the Mass

Conservation Equation and the Momentum Conservation Equation

respectively.

- Energy Conservation Equation to model the compressibility effects.

These equations must be solved in order to obtain the solution of the flow

which is basically the lift, drag, moment, pressure distribution etc. In addition to these

fundamental equations, certain transport equations must be solved to model the

turbulence which is an important characteristic associated with high Reynolds number

flows. The details and derivations of the fundamental and the additional transport

equations can be found in various literatures [28]. Here only the equations are listed

along with a brief explanation.

The Mass Conservation Equation 2.5.1

The equation for conservation of mass, or continuity equation, can be written

as follows:

𝜕𝜌𝜕𝑡

+ ∇ ∙ (𝜌𝐕) = 0

where 𝜌 is the fluid density and 𝐕 is the velocity vector at a point in the flow

field. The above equation is the continuity equation in the form of a partial differential

equation. This equation relates the flow field variables at a point in the flow. It is a

mathematical representation of the law of conservation of mass, i.e., mass can neither

be created nor be destroyed.

The equation physically means that the time rate of change of volume of a

moving fluid element of fixed mass, per unit volume of that element, is equal to the

divergence ∇ of the fluid velocity vector 𝐕. It should be noted here, that the fluid is

assumed to be a continuum rather than a discrete medium.

In two-dimensions, the divergence of the velocity ∇ ∙ 𝐕, is represented

mathematically as follows,

If 𝐕 = 𝑉𝑥𝐢 + 𝑉𝑦𝐣 ≡ 𝑢𝐢 + 𝑣𝐣

Then,

∇ ∙ 𝐕 = 𝜕𝜕𝑥

𝐢 +𝜕𝜕𝑦

𝐣 ∙ (𝑢𝐢 + 𝑣𝐣) = 𝜕𝑢𝜕𝑥

+𝜕𝑣𝜕𝑦

42

Momentum Conservation Equations 2.5.2

For a two-dimensional unsteady viscous flow with density 𝜌, velocities (𝑢, 𝑣)

in Cartesian coordinates (x,y), the Momentum Conservation Equations are given by

the Navier-Stokes equations:

𝜌𝜕𝑢𝜕𝑡

+ 𝜌𝑢𝜕𝑢𝜕𝑥

+ 𝜌𝑣𝜕𝑢𝜕𝑦

= −𝜕𝑝𝜕𝑥

+𝜕𝜕𝑥

λ∇ ∙ 𝐕 + 2𝜇𝜕𝑢𝜕𝑥 +

𝜕𝜕𝑦

𝜇 𝜕𝑣𝜕𝑥

+𝜕𝑢𝜕𝑦

𝜌𝜕𝑣𝜕𝑡

+ 𝜌𝑢𝜕𝑣𝜕𝑥

+ 𝜌𝑣𝜕𝑣𝜕𝑦

= −𝜕𝑝𝜕𝑦

+𝜕𝜕𝑥

𝜇 𝜕𝑣𝜕𝑥

+𝜕𝑢𝜕𝑦 +

𝜕𝜕𝑦

λ∇ ∙ 𝐕 + 2𝜇𝜕𝑣𝜕𝑦

The approximate expression for λ is given as

λ = −23𝜇

The above equations represent the complete Navier-Stokes equations for an

unsteady, two-dimensional viscous flow. However to analyze the compressibility

effects and the turbulence associated with high Reynolds number flows, some

additional equations must be solved. These are the energy and transport equations for

modeling the turbulence.

Viscous Energy Equation 2.5.3

The viscous energy equation is necessary in order to model the compressibility

effects associated with high Reynolds number flows. Compressibility effects are

encountered in gas flows at high velocity and/or in which there are large pressure

variations. When the flow velocity approaches or exceeds the speed of sound of the

gas or when the pressure change in the system is large, the variation of the gas density

with pressure has a significant impact on the flow velocity, pressure, and temperature.

As the Mach number approaches 1.0 (which is referred to as the transonic flow

regime), compressibility effects become important. The energy equation is derived

using the first law of thermodynamics applied to an infinitesimal moving fluid

element, and is mathematically formulated for a two-dimensional viscous flow as

𝜕𝜕𝑡

(𝜌𝐸) = 𝜌 +𝜕𝜕𝑥

𝑘𝜕𝑇𝜕𝑥 +

𝜕𝜕𝑦

𝑘𝜕𝑇𝜕𝑦 − ∇ ∙ 𝑝𝐕

+ 𝜕(𝑢𝜏𝑥𝑥)𝜕𝑥

+ 𝜕𝑢𝜏𝑦𝑥𝜕𝑦

+𝜕𝑣𝜏𝑥𝑦𝜕𝑥

+𝜕𝑣𝜏𝑦𝑦𝜕𝑦

43

Here

- 𝐸 is the total energy on a unit mass basis given by

𝐸 = ℎ −𝑝𝜌

+𝐕2

2

- h is the specific sensible enthalpy given as ℎ = 𝑐𝑝(𝑇 − 𝑇𝑣𝑒𝑓),

𝑇𝑣𝑒𝑓=298.15[K] and 𝑐𝑝 is the specific heat at constant pressure.

- 𝜏𝑥𝑥 and 𝜏𝑦𝑦 are the normal stresses in the x and y directions given by

𝜏𝑥𝑥 = 𝜆(∇ ∙ 𝑉) + 2𝜇𝜕𝑢𝜕𝑥

𝜏𝑦𝑦 = 𝜆(∇ ∙ 𝑉) + 2𝜇𝜕𝑣𝜕𝑦

- 𝜏𝑥𝑦 is the in-plane shear stress given as

𝜏𝑥𝑦 = 𝜏𝑦𝑥 = 𝜇 𝜕𝑣𝜕𝑥

+𝜕𝑢𝜕𝑦

Turbulence Model 2.5.4

Most flows encountered in engineering practice are turbulent and therefore

require different treatment. Turbulent flows are characterized by the following

properties:

- Turbulent flows are highly unsteady,

- They are three-dimensional.

- They contain a great deal of vorticity.

- Turbulence increases the rate at which the conserved quantities are stirred.

- This brings fluids of different momentum content into contact.

- Turbulent flows contain coherent structures—repeatable and essentially

deterministic events that are responsible for a large part of the mixing.

- Turbulent flows fluctuate on a broad range of length and time scales [29].

In order to include the effects of turbulence in any analysis, it is first necessary

to have a model for the turbulence itself. The chief difficulty in modeling turbulent

flows comes from the wide range of length and time scales associated with turbulent

flow. As a result, turbulence models can be classified based on the range of these

length and time scales that are modeled and the range of length and time scales that

44

are resolved. The more turbulent scales that are resolved, the finer the resolution of

the simulation, and therefore the higher the computational cost. If a majority or all of

the turbulent scales are modeled, the computational cost is very low, but the tradeoff

comes in the form of decreased accuracy.

2.5.4.1 Reynolds-averaged Navier–Stokes Equations

One of the widely used approaches in modeling flow turbulence is the

Reynolds-averaged Navier-Stokes (or RANS) equations. In Reynolds-averaged

approaches to turbulence, all of the unsteadiness is averaged out i.e. all unsteadiness is

regarded as part of the turbulence.

In a statistically steady flow, every variable can be written as the sum of a

time-averaged value and a fluctuation about that value:

𝜙(𝑥𝑖 , 𝑡) = 𝜙(𝑥𝑖) + 𝜙′(𝑥𝑖 , 𝑡)

This is known as the Reynolds Decomposition of the variable and forms 𝜙 the

basis of RANS equations. The mean of the variable 𝜙 can be expressed

mathematically as

𝜙(𝑥𝑖) = lim𝑇→∞

1𝑇 𝜙(𝑥𝑖, 𝑡)𝑇

0

This expression is called the time average of the variable 𝜙(𝑥𝑖, 𝑡). Here t is the

time and T is the averaging interval. This interval must be large compared to the

typical scale fluctuations. If T is large enough, 𝜙 does not depend on the time at

which the averaging is started. It must be noted here that 𝜙 is no longer a function of

time and may vary only in space. Therefore all derivatives of 𝜙 should

mathematically be zeros. However, this is often (usually) ignored in modern

treatments of RANS modeling.

Reynolds decomposition possesses the following two properties

𝜙 = 𝜙, and 𝜙′ = 0

The first relation is reflects one of the properties of Reynolds operator R2=R.

The second relation is the property of Reynolds decomposition, according to which

the variable perturbations or fluctuations 𝜙′ are defined such that their time average

equals zero.

45

Based on these properties the equations for mass and momentum

conservations can be written as

𝜕𝜌𝜕𝑡

+ ∇ ∙ (𝜌𝐕) = 0

𝜌 𝑢𝜕𝑢𝜕𝑥

+ 𝜕𝑢𝜕𝑦 = −

𝜕𝜕𝑥

+𝜕𝜕𝑥

λ∇ ∙ 𝐕 + 2𝜇𝜕𝑢𝜕𝑥 +

𝜕𝜕𝑦

𝜇 𝜕𝜕𝑥

+𝜕𝑢𝜕𝑦

−𝜌 𝜕𝑢′2

𝜕𝑥+𝜕𝑢′𝑣′𝜕𝑦

𝜌 𝑢𝜕𝜕𝑥

+ 𝜕𝜕𝑦 = −

𝜕𝜕𝑥

+𝜕𝜕𝑦

λ∇ ∙ 𝐕 + 2𝜇𝜕𝜕𝑦 +

𝜕𝜕𝑥

𝜇 𝜕𝜕𝑥

+𝜕𝑢𝜕𝑦

−𝜌 𝜕𝑢′2

𝜕𝑦+𝜕𝑢′𝑣′

𝜕𝑥

The above equations can be written in the tensor notation as

𝜌𝜕𝑢𝚤𝑢𝚥𝜕𝑥𝑗

= −𝜕𝜕𝑥𝑖

+𝜕𝜕𝑥𝑖

λ∇ ∙ 𝐕 + 2𝜇𝜕𝑢𝚤𝜕𝑥𝑖

+ 𝜇𝜕2𝑢𝚤𝜕𝑥𝑖𝜕𝑥𝑗

− 𝜌𝜕𝑅𝑖𝑗𝜕𝑥𝑖

, 𝑖

= 1, 2 (for 2 − D)

Here the notations can be interpreted as

(𝑢1,𝑢2) ≡ (𝑢 , ), (For the velocities)

(𝑥1, 𝑥2) ≡ (,𝑦), (For the directions)

𝑢𝚤𝑢𝚥 = 𝑢12 𝑢1𝑢2

𝑢2𝑢1 𝑢22 ≡ 𝑢

2 𝑢𝑣𝑣𝑢 2

𝑅𝑖𝑗 is known as the Reynolds stress tensor given by

𝜌𝑅𝑖𝑗 ≡ 𝜌𝑢′𝚤𝑢′𝚥 = 𝜌 𝑢′12

𝑢′1𝑢′2

𝑢′2𝑢′1 𝑢′22 ≡ 𝜌 𝑢′

2 𝑢′𝑣′𝑣′𝑢′ 𝑣′2

As mentioned earlier, in modern RANS modeling of turbulent flows, it is a

common practice to include time derivatives of the mean of the variables, the mass

and momentum equations are formulated as

𝜕𝜌𝜕𝑡

+ ∇ ∙ (𝜌𝐕) = 0

46

𝜕𝜌𝑢𝚤𝜕𝑡

+ 𝜌𝜕𝑢𝚤𝑢𝚥𝜕𝑥𝑗

= −𝜕𝜕𝑥𝑖

+𝜕𝜕𝑥𝑖

λ∇ ∙ 𝐕 + 2𝜇𝜕𝑢𝚤𝜕𝑥𝑖

+ 𝜇𝜕2𝑢𝚤𝜕𝑥𝑖𝜕𝑥𝑗

− 𝜌𝜕𝑅𝑖𝑗𝜕𝑥𝑖

𝐹𝑜𝑟 𝑖 = 1, 2

The above equations are the RANS equations for unsteady flow in two

dimensions. The details of the derivation, terminologies and Reynolds theories can be

found in [30].

2.5.4.2 Boussinesq hypothesis

The Reynolds-averaged approach to turbulence modeling requires that the

Reynolds stresses 𝑅𝑖𝑗 are appropriately modeled. A common method is to employ the

Boussinesq hypothesis.

It was experimentally observed that turbulence decays unless there is shear in

isothermal incompressible flows. Turbulence was found to increase as the mean rate

of deformation increases. Boussinesq proposed in 1877 that the Reynolds stresses

could be linked to the mean rate of deformation. In general, the in plane viscous shear

stresses in two-dimensions are given by

𝜏𝑖𝑗 = 𝜇𝑒𝑖𝑗 = 𝜇 𝜕𝑢𝑖𝜕𝑥𝑗

+𝜕𝑢𝑗𝜕𝑥𝑖

, for 𝑖 = 1,2

On the same lines, linking Reynolds stresses to the mean rate of deformation

(Boussinesq hypothesis), gives

𝜏𝑖𝑗 = −𝜌𝑢′𝚤𝑢′𝚥 = −𝜌𝑅𝑖𝑗 = 𝜇𝑡 𝜕𝑢𝚤𝜕𝑥𝑗

+𝜕𝑢𝚥𝜕𝑥𝑖

Here 𝜇𝑡 is called the turbulent eddy viscosity. It is not homogeneous, i.e. it

varies in space. However it is assumed to be isotropic, i.e. it is the same in all

directions. This assumption is valid for many flows except for flows with strong

separation or swirl. The turbulent viscosity is used to close the momentum equations

and is determined using a turbulence model.

2.5.4.3 Spalart-Allmaras turbulence model

The momentum equations can be closed by solving a turbulence model

equation. There are a number of turbulence models that are in use in modern

turbulence analysis such as the Spalart-Allmaras model, 𝑘 − 𝜀 models and the 𝑘 − 𝜔

47

models. Here the Spalart-Allmaras model will be used. This turbulence model, which

is a one-equation model as opposed to the others, can be used to calculate the dynamic

eddy viscosity, 𝜇𝑡, by solving the one-equation transport model for 𝜈 which is

identical to the turbulent kinematic viscosity. The transport equation is given by

𝜕𝜕𝑡

(𝜌𝜈) +𝜕𝜕𝑥𝑖

(𝜌𝜈𝑢𝑖) = 𝐺𝜈 +1𝜎𝜈𝜕𝜕𝑥𝑗

(𝜇 + 𝜌𝜈)𝜕𝜈𝜕𝑥𝑗

+ 𝐶𝑏2𝜌 𝜕𝜈𝜕𝑥𝑗

2

− 𝑌𝜈

Here 𝐺𝜈 is the production of turbulent viscosity, and 𝑌𝜈 is the destruction of

turbulent viscosity that occurs in the near-wall region due to the wall blocking and

viscous damping. 𝜎𝜈 and 𝐶𝑏2 are the constants and 𝜈 is the molecular kinematic

viscosity[31].

The turbulent viscosity, 𝜇𝑡, is computed from

𝜇𝑡 = 𝜌𝜈𝑓𝜈1

where the viscous damping function, 𝑓𝜈1, is given by

𝑓𝜈1 =𝑋3

𝑋3 + 𝐶𝜈13 and 𝑋 ≡

𝜈𝜈

, Here 𝐶𝜈1 is a constant.

The turbulent viscosity production term 𝐺𝜈 is modeled as

𝐺𝜈 = 𝐶𝑏1𝜌𝜈

where

≡ 𝑆 +𝜈

𝜅2𝑑2𝑓𝜈2 and 𝑓𝜈2 = 1 −

𝑋1 + 𝑋𝑓𝜈1

𝐶𝑏1 and 𝜅 are constants, 𝑑 is the distance from the wall, and 𝑆 is a scalar

measure of the deformation tensor. 𝑆 is based on the magnitude of the vorticity:

𝑆 ≡ 2Ω𝑖𝑗Ω𝑖𝑗 where Ω𝑖𝑗 is the mean rate-of-rotation tensor and is defined by

Ω𝑖𝑗 =12𝜕𝑢𝑖𝜕𝑥𝑗

−𝜕𝑢𝑗𝜕𝑥𝑖

The turbulent viscosity destruction term is modeled as

𝑌𝜈 = 𝐶𝑤1𝜌𝑓𝑤 𝜈𝑑2

𝑤ℎ𝑒𝑟𝑒 𝑓𝑤 = 𝑔 1 + 𝐶𝑤36

𝑔6 + 𝐶𝑤36

16

48

𝑔 = 𝑟 + 𝐶𝑤2(𝑟6 − 𝑟) , 𝑤ℎ𝑒𝑟𝑒 𝑟 =𝜈

𝜅2𝑑2

Table 1 Model Constants

2.6 Computational Fluid Dynamics

Computational fluid dynamics (CFD) is the science of predicting fluid flow,

heat transfer, mass transfer, chemical reactions, and related phenomena by solving the

mathematical equations which govern these processes using a numerical process.

The governing equations, such as the mass, momentum and energy

conservation equations, and the turbulence models, described in the previous section

cannot be solved analytically except in special cases. Thus, the solution to these

equations is obtained numerically. In order to approximate the solution, a

discretization method is used which approximates the differential equations by a set of

algebraic equations, which can then be solved simultaneously or iteratively.

Broadly, the strategy of CFD is to replace the continuous problem domain

with a discrete domain using a grid. In the continuous domain, each flow variable is

defined at every point in the domain. For instance, the pressure p in the continuous 1D

domain shown in the figure below would be given as

𝑝 = 𝑝(𝑥), 0 < 𝑥 < 1

𝐶𝑏1 0.1355

𝐶𝑏2 0.622

𝜎𝜈 2/3

𝐶𝜈1 7.1

𝐶𝑤1 𝐶𝑏1𝜅2

+(1 + 𝐶𝑏2)

𝜎𝜈

𝐶𝑤2 0.3

𝐶𝑤3 2.0

𝜅 0.4187

49

x=0 x=1 x1 xN xi

Continuous Domain Discrete Domain

0 < x < 1 x = x1, x2, …, xN

Coupled PDEs+Boundary

conditions in continuous variables Coupled algebraic equations

in discrete variables

Figure 9 The Computational Spatial Grid

In the discrete domain, each flow variable is defined only at the grid points.

So, in the discrete domain shown below, the pressure would be defined only at the N

grid points. 𝑝𝑖 = 𝑝(𝑥𝑖), 𝑖 = 1,2, … ,𝑁

Discretization 2.6.1

In a CFD solution, relevant flow variables are solved only at the grid points.

The values at other locations are determined by interpolating the values at the grid

points. The governing partial differential equations and boundary conditions are

defined in terms of the continuous variables 𝑝,𝐕etc in theoretical fluid dynamics,

which are essentially functions of space and time. These can be approximate in the

discrete domain in terms of the discrete variables 𝑝𝑖,𝐕𝑖etc. The discrete system is a

large set of coupled, algebraic equations in terms of discrete variables.

One of the many approaches to numerically solving the governing equations

of flow is the Finite Volume method. This method forms the basis of almost 80

percent of present day flow solvers. In the finite-volume approach, the solution

domain is subdivided into a finite number of small control volumes (cells) by a grid.

Figure 8 Continuous Domain and Discrete Domain

50

This sub-division is carried out using meshing algorithms. The grid defines the

boundaries of the control volumes while the computational node lies at the center of

the control volume. The integral form of the conservation equations are applied to the

control volume defined by a cell to get the discrete equations for the cell. For

example, the integral form of the continuity equation for steady, incompressible flow

is

𝜌𝐕 ∙ 𝐝𝐬 = 0𝑆

The integration is over the surface S of the control volume and 𝐝𝐬 is the

elemental area vector. Physically, this equation means that the net volume flow into

the control volume is zero. For any rectangular cell from the grid as shown below

The velocity at face i is taken to be 𝐕𝑖 = 𝑢𝑖𝐢 + 𝑣𝑖𝐣. Applying the mass

conservation equation to the control volume defined by the cell gives

−𝑢1∆𝑦 − 𝑣2∆𝑥 + 𝑢3∆𝑦 + 𝑣4∆𝑥 = 0

This is the discrete form of the continuity equation for the cell. It is equivalent

to summing up the net mass flow into the control volume and setting it to zero. So it

ensures that the net mass flow into the cell is zero i.e. that mass is conserved for the

cell. Usually, though not always, the values at the cell centers are solved for directly

by inverting the discrete system. The face values 𝑢1, 𝑣2, etc. are obtained by suitably

interpolating the cell-center values at adjacent cells.

Figure 10 Typical finite volume cell

51

In a general situation, the discrete equations are applied to the cells in the

interior of the domain. For cells at or near the boundary, a combination of the discrete

equations and boundary conditions would be applied. In the end, one would obtain a

system of simultaneous algebraic equations with the number of equations being equal

to the number of independent discrete variables. This set can then be solved to obtain

the values of the variables at their respective grid points.

CFD based Shape Optimization 2.6.2

The focus of CFD applications has shifted from mere analysis to aerodynamic

shape design. This shift has been mainly motivated by the availability of high

performance computing platforms and by the development of new and efficient

analysis and design algorithms. In particular automatic design procedures, which use

CFD combined with gradient-based optimization techniques, have had a significant

impact on the design process by removing difficulties in the decision making process

faced by the aerodynamicist.

With recent research and code development efforts in the area of

computational fluid dynamics, CFD has proven to be useful in supporting product

design and development in many industrial applications. For many product designs

where fluid flow simulations are needed, CFD analyses have proven to be quite useful

in predicting the flow pattern for a given set of design parameters [32].

Aerodynamic shape optimization strategies integrated with CFD usually come

into play at the preliminary design phase of an airplane. It is here that the conceptual

design is refined from a shape optimization point of view. Modern shape optimization

strategies usually involve the integration of a CFD code with an optimization

algorithm. The CFD code performs the flow analysis on a specific shape and

provides the optimization algorithm with values of the required components that make

up the objective function. The optimization algorithm, based on the evaluation of the

objective function value, perturbs the geometry in a direction of decreasing objective

function gradients. The new geometry is then re-analyzed and the process is repeated

till an optimum shape is reached.

52

3. ITERATIVE RESPONSE SURFACE BASED OPTIMIZATION

3.1 Introduction

The evaluation of aerospace designs is synonymous with the use of long

running and computationally intensive simulations. Unlike the engineering

methodologies adopted in the middle of the 20th century to design aerospace systems,

in which predominantly hand calculations and wind tunnel tests were used in a cut-

and-try fashion, engineers in the late 20th century have resorted to Computer Aided

Engineering (CAE) of which CFD is an important part.

CFD, as outlined in the previous chapter, has evolved from a mere flow

analysis tool to an important design tool. However, even with the ever expanding

computational resources, it has widely been regarded as a computationally expensive

platform, especially when it comes to very high-fidelity flow simulations. The use of

long running expensive computer simulation in design, therefore leads to a

fundamental problem when trying to compare and contrast various competing

options— there are never sufficient resources to analyze all of the combinations of

variables that one would wish [33].

This problem is particularly acute when using optimization schemes. All

optimization methods depend on some form of internal model of the problem space

they are exploring— for example a quasi-Newton scheme attempts to construct the

Hessian at the current design point by sampling the design space. To build such a

model when there are many variables can require large numbers of analyses to be

carried out, particularly if using finite difference methods to evaluate gradients.

Objective function and constraints in aerodynamic shape optimization

involving transonic flow numerical simulation, such as CFD, may be non-smooth and

noisy. Non-smoothness is created by the presence of flow discontinuities such as

shock waves. Noise can be caused either by the changes in computational mesh

geometry due to free boundaries or by poor convergence of numerical schemes.

Although these features can make a small change in some design parameters, it could

lead to a huge ramification in the objective function or constraints.

These non-smoothness and noise issues of the objective function become more

serious in gradient-based optimization methods (GBOMs), where the objective

53

function value as well as its gradient information is used. In multidisciplinary design

optimization (MDO) problems, which usually have objective functions coupled with

numerous constraints, it is significantly difficult to formulate the design problem with

GBOMs. Because the optimization depends greatly on the formulation of the design

problem, the process of searching for the optimum is likely to render just a local

value. Another shortcoming of GBOMs is that because many analysis programs were

not written with an automated design in mind, adaptation of these programs to an

optimization code may need significant reprogramming in the analysis routine [34].

3.2 Response Surface Optimization

In statistics, response surface methodology explores the relationships between

several explanatory variables and one or more response variables. The method was

introduced by G. E. P. Box and K. B. Wilson in 1951. The main idea of response

surface methodology is to use a sequence of designed experiments to obtain an

optimal response. Incorporating this methodology in design optimization falls into the

category of Surrogate or Response Surface Optimization (RSO). It has been shown to

be an effective approach for the design of computationally expensive models such as

those found in aerospace systems, involving aerodynamics, structures, and propulsion,

among other disciplines. Successful applications include the multidisciplinary optimal

design of aerospike rocket nozzles, injectors and turbines for liquid rocket propulsion,

and supersonic business aircrafts [35].

For a new or a computationally expensive design, optimization based on an

inexpensive surrogate, such as Response Surface Model (also known as surrogate or

approximation models), is a good choice. RSO allows for the determination of an

optimum design, while at the same time providing insight into the workings of the

design. A response model not only provides the benefit of low-cost for function

evaluations, but it can also help revise the problem definition of a design task, which

is not unusual for new efforts. Furthermore, it can conveniently handle the existence

of multiple desirable design points and offer quantitative assessments of trade-offs as

well as facilitate global sensitivity evaluations of the design variables[36].

Thus, the use of Response Surface Models (RSM) in optimization is becoming

increasingly popular. The RSM is not in itself an optimizer, but rather a tool for

increasing the speed of optimization. Instead of making direct calls to an expensive

54

numerical analysis code, such as CFD, an optimization routine takes values from a

cheap surrogate model, that is formulated using a specific set of responses obtained

from the numerical code. The popularity of such methods has probably increased due

to the development of approximation methods which are better able to capture the

nature of a multi-modal design space.

The main objective behind creating an RSM is to be able to predict the

response of a system for an operating point without actually performing a simulated

analysis at that point. The response of the system can then be predicted just by

inputting the operating point values into the RSM and obtaining the value of the

response. The RSM basically takes the shape of a mathematical equation 𝑓(𝐱),

essentially a quadratic polynomial, which takes the values of the design variables X as

an input, and returns an approximated value of the system response. Various

optimization methodologies can then be employed to optimize this computationally

cheap response model in order to obtain the best operating point. Some of the other

benefits of using RSM include—

- It smoothens out the high-frequency noise of the objective function and is,

thus, expected to find a solution near the global optimum.

- Various objectives and constraints can be attempted in the design process

without additional numerical computations.

- It does not require a modification in analysis codes [1].

RSO is composed of four phases:

- Sampling (Design of Experiments)—this basically involves testing or

obtaining actual values of the system response, by performing simulations

for a select set of points within the design space.

- Response Surface Construction—based on the responses obtained for the

sampling points, a RSM is constructed. The RSM is an approximation of

the system response.

- RSO—Optimization algorithms are used to optimize the RSM and obtain

the best operating point values of the system.

- RSM improvement—The RSM approximation is improved by training it

further by including additional simulated responses.

55

Geometry Parameterization

Design of Experiments

CFD CFD CFD

Construct RSM

Search RSM

Design Adequate?

OPIMUM DESIGN

RSM

impr

ovem

ent

NO

YES

CFD

Design Space 3.2.1

From an aerodynamic shape optimization point of view, the system is basically

the airfoil geometry that has to be optimized for a specific operating condition

(airspeed and altitude). The design points are the design variables that completely

define the airfoil geometry. As discussed in chapter 2, the design variables constitute

the y-ordinates of the 10 B-spline control points that control the shape of the upper

and lower surfaces of the airfoil, and the angle of attack.

The design space is the region bounded by the upper and lower limits of the

design variables. This implies that the design variables are allowed to vary only

Figure 11 General RSO procedure

56

within the limits defined by the design space. It is defined such that, overly unusual or

unrealistic shapes are not attained. It is also dependent on the structural and material

constraints. For example, for any variant of the airfoil shape, the skin should not

deform beyond the limits of elasticity, or the thickness should not be less than the

required minimum thickness in-order to have enough room for the fuel tank etc.

Design of Experiments 3.2.2

In the Design of Experiments (DoE) phase the design space is systematically

explored using a technique, which generates the test matrix of design points to be

probed in each computational experiment. The objective of this task is to fragment the

design space in a specific format such that a matrix of design variable values is

obtained. This is done by discretizing the variation range of each design variable into

𝑁𝑠 levels. Combining the values of all the design variables at a specific level yields

one experiment. Combining all the experiments therefore forms a set of 𝑁𝑠

experiments, which is thereby referred to as a DoE.

If X is the design vector consisting of Nvar design variables (DV), and if each

design variable is split into 𝑁𝑠 levels, the DoE matrix is given by,

𝐗𝐷𝐷𝐷 =

𝐱11 𝐱12 ⋯𝐱21 𝐱22 ⋯

𝐱1𝑁var𝐱2𝑁var

⋮ ⋮ ⋱ 𝐱𝑁𝑠1 𝐱𝑁𝑠2 ⋯

⋮𝐱𝑁𝑠𝑁var

𝐷𝑒𝑠𝑖𝑔𝑛 𝑉𝑣𝑣𝑖𝑣𝑏𝑙𝑒𝑠 𝐷𝑉 1 𝑡𝐷 𝐷𝑉 𝑁var

← 𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 − 1← 𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 − 2

⋮← 𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 − 𝑁𝑠

3.2.2.1 Latin Hypercube Sampling

Latin Hypercube Sampling (LHS) is a statistical method for generating a

distribution of plausible collections of design variable values from a multidimensional

distribution. This method is often used as a DoE technique.

In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and

a cube (n = 3). It is a closed, compact, convex figure whose 1-skeleton consists of

groups of opposite parallel segments aligned in each of the

space's dimensions, perpendicular to each other and of the same length.

A hyperplane is also a concept which is a generalization of the plane into a different

number of dimensions. A hyperplane of an n-dimensional space is a flat subset with

dimension n − 1.

57

In the context of statistical sampling, a square grid containing sample positions

is a Latin square if (and only if) there is only one sample in each row and each

column. A Latin Hypercube is the generalization of this concept to an arbitrary

number of dimensions, whereby each sample is the only one in each axis-aligned

hyperplane containing it.

When sampling a function of 𝑁𝑣𝑣𝑣 variables, the range of each variable is

divided into equally probable intervals. The 𝑁𝑠 sample points are then placed to

satisfy the Latin Hypercube requirements; doing so forces the number of divisions,𝑁𝑠,

to be equal for each variable. It should be noted that this sampling scheme does not

require more samples for more dimensions (variables); this independence is one of the

main advantages of this sampling scheme. Another advantage is that random samples

can be taken one at a time, remembering which samples were taken so far. For

example, for 𝑁𝑣𝑣𝑣 = 4 (4 design variables), and 𝑁𝑠 = 4 (4 levels), a Latin Hypercube

Sampling may take the form:

Building a Latin hypercube, that is the multidimensional, can be done in a

similar way. The design space of each dimension is split into equal number of levels

and the points are placed in the levels such that any arbitrary vector emerging from

the points in a direction parallel to any of the dimensional axes does not encounter

with any other point in its way.

This is achieved using the following technique. If X denotes the 𝑁𝑠 × 𝑁𝑣𝑣𝑣 the

DoE matrix 𝑁𝑠 points in 𝑁𝑣𝑣𝑣 dimensions (each row represents a point), then each

column of X is filled with random permutations (1, 2, . . . , 𝑁𝑠 ) and stratified such

that no specific point in any row is repeated in more than one column. This set is then

normalized such that values lie within [0,1]𝑁𝑣𝑎𝑟 .

2 1 3 4 ← Level − 1

3 2 4 1 ← Level − 2

1 4 2 3 ← Level − 3

4 3 1 2 ← Level − 4

↑ DV 1 ↑ DV 2 ↑ DV 3 ↑ DV 4

58

3.2.2.2 Space Filling LHS

One of the most widely-used measures to evaluate the uniformity (space-

fillingness) of a sampling plan is the maximin metric introduced by Johnson et al.

(1990). The criterion based on this may be defined as follows—

Let 𝑑1,𝑑2, … , 𝑑𝑚 be the list of the unique values of distances between all

possible pairs of points in a DoE X, sorted in the ascending order. Here the distance

𝑑𝑗 is basically defined by the p-norm of the space given by:

Figure 12 Three-variable, ten-point Latin hypercube sampling plan shown in three dimensions (top left), along with its two-dimensional projections.

59

𝑑𝑝𝐱(𝑖1), 𝐱(𝑖2) = 𝐱(𝑖1) − 𝐱(𝑖2)𝑝

𝑘

𝑗=1

1 𝑝

For 𝑝 = 1 this is the rectangular distance and 𝑝 = 2 yields the Euclidean

norm. There is little evidence in the literature of one being more suitable than the

other for sampling plan evaluation if no assumptions are made regarding the structure

of the model to be fitted, though it must be noted that the rectangular distance is

considerably cheaper to compute.

Further, let 𝐽1, 𝐽2, … , 𝐽𝑚 be defined such that 𝐽𝑗 is the number of pairs of points

in X separated by the distance 𝑑𝑗—Then X is called a maximin plan among all

available plans if it maximizes 𝑑𝑗 and, among plans for which this is true, minimizes

𝐽𝑗 [37].

3.2.2.3 Design Matrix

The design matrix is formed by concatenating the values of the design

variables at all levels. In order to do so, the design space needs to be discretized into

levels which are equal to the desired number of computer simulations to be

performed. The design space as described above is the region bounded by the upper

and lower limits of the design variables. These are the y-ordinates of the 10 control

points (five control points for each surface) and the angle of attack. CPs 2 through 6

control the upper surface while CPs 9 through 13 control the lower surface of the

airfoil.

For this, the lower and upper limits for each of the 10 control points and the

angle of attack are defined. The lower limit of the control points is taken to be 75% of

the baseline values and the upper limit is taken to be 25% above the baseline values.

The lower and upper limits of the angle of attack are case dependent varying between

0 and 4o.

The range of each of the design variables DVRange (the design space) is the

difference between the upper, DVUpper, and lower limits, DVLower, of the design

variable. This range is discretized into equal number of levels 𝑁𝑠 which is equivalent

to the number of experiments (computer simulations) to be performed. To obtain the

values of the design variables at each level, first a LHS plan is generated for the 11

60

design variables and 𝑁𝑠 levels. This generates a matrix L of size (𝑁𝑠x 11), with the 𝑁𝑠

values in each of the 11 columns varying from 0 to 1 in a LHS pattern.

The values of the design variables at each level are then obtained based on the

following equation:

𝐗𝐷𝐷𝐷(𝑖, 𝑗) = 𝐷𝑉Lower(𝑗) + 𝐷𝑉Range(𝑗) × 𝐿(𝑖, 𝑗) , For 𝑖 = 1, 2, … ,𝑁𝑠

For 𝑗 = 1, 2, … , 11

Figure 13 Lower Limit, Upper limit and Baseline Control Points and Airfoil shapes

61

The matrix thus formed, describes the set of airfoil geometries for which the

CFD simulations are to be performed in order to construct the RSM.

Constructing the RSM 3.2.3

RSM builds a response model by calculating data points with experimental

design theory to prescribe a response of a system with independent variables. The

relationship can be written in a general form as follows:

𝑦 = 𝐹(𝐗) + ε

where ε represents the total error, which is often assumed to have a normal

distribution with a zero mean. Consider a sampling plan 𝐗 and a set of 𝑁𝑠 observed

values comprising the responses obtained from the computer simulations:

𝐗 = 𝐗𝐷𝐷𝐷 =

𝑥11 𝑥12 ⋯𝑥21 𝑥22 ⋯

𝑥1𝑁var𝑥2𝑁var

⋮ ⋮ ⋱ 𝑥𝑁𝑠1 𝑥𝑁𝑠2 ⋯

⋮𝑥𝑁𝑠𝑁var

⟶⟶⋮⟶

𝐲𝟏𝐲𝟐⋮𝐲𝑁𝑠

The polynomial approximation of order m (degree 𝑚 − 1) of an underlying

function f is, essentially, a Taylor series expansion of f truncated after 𝑚 − 1 terms.

This suggests that a higher order expansion will usually yield a more accurate

approximation. However, the greater the number of terms, the more flexible the

model becomes and there is a danger of over-fitting the noise that may be corrupting

the underlying response. For this reason, the order of the polynomial has been

restricted to 3.

A full quadratic polynomial (degree 2, order 3) approximation of F can be

written as:

𝐲 = 𝑓(𝐱,𝜷) = 𝛽1 + 𝛽𝑖𝑥𝑖

𝑁var

𝑖=1

+ 𝛽𝑗𝑗

𝑁var

𝑗=1

𝑥𝑗2 + 𝛽𝑖𝑗𝑥𝑖𝑥𝑗

𝑁var

𝑗=i+1

𝑁var−1

𝑖=1

Here 𝛽0,𝛽𝑖,𝛽𝑖𝑗 etc. are the regression coefficients of the polynomial. The total

number of these coefficients is 𝑛𝑡 = (𝑁var + 1)(𝑁var + 2)/2. These values can be

determined using the standard least-square fitting regression of an over determined

problem:

𝐲 = 𝚽𝛃

62

Here y is the initial response matrix 𝑦1,𝑦2, … ,𝑦𝑁𝑠𝑇and 𝚽 is the

Vandermonde matrix of size (𝑁𝑠 × 𝑁var) given by:

𝚽 =

⎣⎢⎢⎢⎡11⋮1

𝑥11𝑥21⋮

𝑥𝑁𝑠1

𝑥12𝑥22⋮

𝑥𝑁𝑠2

…………

𝑥1𝑁var 𝑥2𝑁var

⋮𝑥𝑁𝑠𝑁var

𝑥112

𝑥212⋮

𝑥𝑁𝑠12

𝑥122

𝑥222⋮

𝑥𝑁𝑠22

…………

𝑥1𝑁var2

𝑥2𝑁var2

⋮𝑥𝑁𝑠𝑁var

2

𝑥11𝑥12𝑥21𝑥22⋮

𝑥𝑁𝑠1𝑥𝑁𝑠2

𝑥11𝑥13𝑥21𝑥23⋮

𝑥𝑁𝑠1𝑥𝑁𝑠3

…………

𝑥1𝑁var−1𝑥1𝑁var𝑥2𝑁var−1𝑥2𝑁var

⋮ 𝑥𝑁𝑠𝑁var−1𝑥𝑁𝑠𝑁var⎦

⎥⎥⎥⎤

And 𝛃 =

𝛽1𝛽2⋮𝛽𝑛𝑡

3.2.3.1 Linear Least-Squares Solution

The method of least squares is a standard approach to the approximate solution

of over determined systems, i.e., sets of equations in which there are more equations

than unknowns. Least-squares means that the overall solution minimizes the sum of

the squares of the errors made in the results of every single equation.

Such a system, as described above, usually has no solution, so the goal is

instead to find the coefficients 𝛃 which fit the equations best, in the sense of solving

the quadratic minimization problem:

𝛃 = min𝛃𝑆(𝛃)

Here the objective function 𝑆 is given by

𝑆(𝛃) = 𝑦𝑖 −Φ𝑖𝑗β𝑗

𝑛𝑡

𝑗=1

2

=𝑁𝑠

𝑖=1

‖𝐲 − 𝚽𝛃‖2

Considering the ith residual to be

𝑟𝑖 = 𝑦𝑖 −Φ𝑖𝑗β𝑗

𝑛𝑡

𝑗=1

Then 𝑆 can be written as

𝑆 = 𝑟𝑖2𝑁𝑠

𝑖=1

.

63

𝑆 is minimized when the gradient vector is zero. The elements of the gradient

vector are the partial derivatives of 𝑆 with respect to the parameters.

𝜕𝑆𝜕β𝑗

= 2𝑟𝑖

𝑁𝑠

𝑖=1

𝜕𝑟𝑖𝜕β𝑗

, 𝑓𝑜𝑟(𝑗 = 1, 2, … ,𝑛𝑡)

And 𝜕𝑟𝑖𝜕β𝑗

= −Φ𝑖𝑗

Substitution of the expressions for the residuals and the derivatives into the

gradient equations gives

𝜕𝑆𝜕β𝑗

= 2𝑦𝑖 − X𝑖𝑘β𝑘

𝑛𝑡

𝑘=1

−X𝑖𝑗, 𝑓𝑜𝑟 (𝑗 = 1, 2, … ,𝑛𝑡)𝑁𝑠

𝑖=1

Thus, if β minimizes 𝑆, then

2𝑦𝑖 −Φ𝑖𝑘β𝑘

𝑛𝑡

𝑘=1

−Φ𝑖𝑗 = 0, 𝑓𝑜𝑟 (𝑗 = 1, 2, … ,𝑛𝑡)𝑁𝑠

𝑖=1

Upon rearrangement,

Φ𝑖𝑗Φ𝑖𝑘β𝑘

𝑛𝑡

𝑘=1

𝑁𝑠

𝑖=1

= Φ𝑖𝑗𝑦𝑖

𝑛𝑡

𝑘=1

, 𝑓𝑜𝑟 (𝑗 = 1, 2, … ,𝑛𝑡)

The normal equations are written in the matrix notation as

(𝚽𝑶𝚽)𝛃 = 𝚽𝑶𝐲,

or 𝛃 = (𝚽𝑶𝚽)−𝟏𝚽𝑶𝐲

The solution of the normal equations yields the vector 𝛃 of the optimal

parameter values. It must be noted that in order to construct a sufficiently trained

RSM, the number of samples (or levels) 𝑁𝑠 must be 1.5~3 times the number of

regression coefficients, 𝑛𝑡 .

3.2.3.2 Model Testing

Once the RSM is available, it is imperative to establish the predictive

capabilities of the surrogate model away from the available data. In the context of an

RSA, several measures of predictive capability are available

64

Adjusted root mean square error

The error ε𝑖 at any point is i is given by

ε𝑖 = 𝑦𝑖 − 𝑦𝑖

where 𝑦𝑖 is the actual value and 𝑦𝑖 is the predicted value. Hence the adjusted

root mean square (RMSadj) error σ𝑣 is given by

σ𝑣 = ∑ ε𝑖2𝑁𝑠𝑖=1

(𝑁𝑠 − 𝑛𝑡)

For a good fit, σ𝑣 should be small compared to the data.

Coefficient of multiple determination

The adjusted coefficient of multiple determination 𝑅𝑣𝑑𝑗2 defines the prediction

capability of the RSM as

𝑅𝑣𝑑𝑗2 = 1 − σ𝑣2(𝑁𝑠 − 1)∑ (𝑦𝑖 − 𝑦)2𝑁𝑠𝑖=1

, where 𝑦 =∑ 𝑦𝑖𝑁𝑠𝑖=1𝑁𝑠

For a good fit, 𝑅𝑣𝑑𝑗2 should be close to 1.

3.2.3.3 The RSM

From the aerodynamic shape optimization perspective with 11 design variables

(𝑁var = 11), the number of coefficients are 𝑛𝑡 = 78. A typical RSM would take the

form—

𝑦 ≡ 𝑓(𝐱,𝜷) =

𝛽1 + 𝛽2𝑥1 + 𝛽3𝑥2 + 𝛽4𝑥3 + 𝛽5𝑥4 + 𝛽6𝑥5 + 𝛽7𝑥6 + 𝛽8𝑥7 + 𝛽9𝑥8 + 𝛽10𝑥9 + 𝛽11𝑥10 +

𝛽12𝑥11 + 𝛽13𝑥12 + 𝛽14𝑥22 + 𝛽15𝑥32 + 𝛽16𝑥42 + 𝛽17𝑥52 + 𝛽18𝑥62 + 𝛽19𝑥72 +

𝛽20𝑥82 + 𝛽21𝑥92 + 𝛽22𝑥102 + 𝛽23𝑥112 +

𝛽24𝑥1𝑥2 + 𝛽25𝑥1𝑥3 + 𝛽26𝑥1𝑥4 + 𝛽27𝑥1𝑥5 + 𝛽28𝑥1𝑥6 + 𝛽29𝑥1𝑥7 + 𝛽30𝑥1𝑥8 +

𝛽31𝑥1𝑥9 + 𝛽32𝑥1𝑥10 + 𝛽33𝑥1𝑥11 +

𝛽34𝑥2𝑥3 + 𝛽35𝑥2𝑥4 + 𝛽36𝑥2𝑥5 + 𝛽37𝑥2𝑥6 + 𝛽38𝑥2𝑥7 + 𝛽39𝑥2𝑥8 + 𝛽40𝑥2𝑥9

+𝛽41𝑥2𝑥10 + 𝛽42𝑥2𝑥11 +

𝛽43𝑥3𝑥4 + 𝛽44𝑥3𝑥5 + 𝛽45𝑥3𝑥6 + 𝛽46𝑥3𝑥7 + 𝛽47𝑥3𝑥8 + 𝛽48𝑥3𝑥9 + 𝛽49𝑥3𝑥10

+𝛽50𝑥3𝑥11

+𝛽51𝑥4𝑥5 + 𝛽52𝑥4𝑥6 + 𝛽53𝑥4𝑥7 + 𝛽54𝑥4𝑥8 + 𝛽55𝑥4𝑥9 + 𝛽56𝑥4𝑥10 + 𝛽57𝑥4𝑥11 + ⋯

65

𝛽58𝑥5𝑥6 + 𝛽59𝑥5𝑥7 + 𝛽60𝑥5𝑥8 + 𝛽61𝑥5𝑥9 + 𝛽62𝑥5𝑥10 + 𝛽63𝑥5𝑥11 +

𝛽64𝑥6𝑥7 + 𝛽65𝑥6𝑥8 + 𝛽66𝑥6𝑥9 + 𝛽67𝑥6𝑥10 + 𝛽68𝑥6𝑥11 +

𝛽69𝑥7𝑥8 + 𝛽70𝑥7𝑥9 + 𝛽71𝑥7𝑥10 + 𝛽72𝑥7𝑥11 +

𝛽73𝑥8𝑥9 + 𝛽74𝑥8𝑥10 + 𝛽75𝑥8𝑥11 +

𝛽76𝑥9𝑥10 + 𝛽77𝑥9𝑥11 +

𝛽78𝑥10𝑥11

Optimizing the RSM 3.2.4

Optimizing the RSM is the third step in the response surface optimization

process. The RSM obtained, 𝐲 ≡ 𝑓(𝐗,𝛃), as described earlier is a quadratic

polynomial consisting of 𝑛𝑡 terms in 𝑁var dimensions (design variables). It is

basically an algebraic equation which takes the values of the design variables (𝐗) and

parameters (𝛃) as the input and returns a scalar value which is an approximation of

the system’s response.

Also, as the RSM is a quadratic polynomial, it is expected to be a smooth

continuous function in terms of the design variables 𝐗. Therefore there are no issues

relating to noise and non-smoothness that are associated with direct optimization

method. Optimizing the RSM is therefore simpler, computationally less expensive and

efficient when compared to direct optimization of the system (i.e. optimizing using

the gradient values of the system response).

The optimization is carried on the RSM using a Sequential Quadratic

Programming (SQP) algorithm. This algorithm has been described at length in chapter

2. The values of the design variables obtained at the end of the SQP implementation,

(𝐗∗), are the values that minimize the RSM. These are the optimum values of the

design variables that minimize the objective function.

System Response 3.2.5

The system response, a term which has been excessively used in the

descriptions above, is the scalar value of the objective functions that are defined for

the purpose of aerodynamic shape optimization.

The objective function used here is the Lift-Moment Constrained Drag

minimization problem given by

66

𝐽 =

⎩⎪⎨

⎪⎧𝜔𝐿 1 −

𝐶𝐿𝐶𝐿∗2

+ 𝜔𝑀 1 −𝐶𝑀𝐶𝑀∗2

+ 𝜔𝐷 1 −𝐶𝐷𝐶𝐷∗2

if 𝐶𝐷 > 𝐶𝐷∗

𝜔𝐿 1 −𝐶𝐿𝐶𝐿∗2

+ 𝜔𝑀 1 −𝐶𝑀𝐶𝑀∗2

otherwise

Here 𝜔𝐿 ,𝜔𝑀,𝜔𝐷are user defined weights, and 𝐶𝐿∗,𝐶𝑀∗ ,𝐶𝐷∗ are the target

(required) lift, moment and drag coefficients respectively. This function has been

described at length in Chapter 2 (section 2.4). The above function is evaluated using

the results obtained from the CFD solver. The solver performs the flow simulation for

an airfoil shape that is obtained from the set of design variables (Control point

coordinates and angle of attack) for a particular operating condition (Mach number,

operating pressure, temperature).

At the end of the simulation, the CFD solver yields the values of

𝐶𝐿 ,𝐶𝑀 and 𝐶𝐷 for the shape being investigated. These values are then used to evaluate

the objective function as above. This value of 𝐽 is considered as a particular response

of the system at a specific operating point (in this case, the operating point is the

above airfoil shape governed by specific values of the design variables). A set

consisting of multiple values of such operating points, in conjunction with the

corresponding values of their responses, is used to construct the RSM as described in

the previous sections.

3.3 Iterative Improvement of the RSM

In the standard RSO scheme the design space is normally explored using a

space filling DoE technique. The range of variation of the design variables is chosen

about a reference design, considering a large portion of the design space. On these

selected points, the objective functions are evaluated using the CFD solver and the

information is used to build a RSM based on a least-square-fitting quadratic

polynomial. An optimization problem based on the RSM is then solved using a

gradient method, such as SQP as described above, to find the minimum.

Because these approximations carry a bias error, the minimum thus found

needs to be validated. Therefore an additional CFD run is performed at the end of the

process to verify the performance of the optimum design obtained by solving the

equivalent optimization problem.

67

To improve the prediction capability of the RSM, an iterative RSO process

can be employed. This process can be summarized as follows—

1) An initial optimum is found solving the constrained minimization problem

based on the least-square fitting quadratic polynomials of the objective and

constraint functions, built upon a large design space.

2) The actual value of the objective function is computed for the optimum

design found. This computation is done based on the values obtained from

a CFD solver run for the optimum airfoil shape found in step 1.

3) The relevant geometry values are added to the initial set of samples (the

initial set used to construct the RSM). The actual value of the objective

function (found from step 2) is added to the set of responses.

4) Least-square fitting and minimization are performed again on the basis of

the new set of data.

5) Steps 2 to 4 are repeated until the minimum computed from the response

surface optimization and the function evaluation does not vary within a

specified tolerance [38].

When no further cost function reduction is obtained, an additional step can be

performed. The bounds of the design space of the SQP optimization based on the

polynomial approximation can be changed, by updating the center point with the

position of the last optimum found, while maintaining its size. To approximate the

actual function in this new region of the design space, the polynomial computed as the

least-square fit of all the previous design points are used and iteratively corrected

about the current minimum in the same way as it is done previously.

Algorithm Flow Chart 3.3.1

The algorithm begins with an initial airfoil geometry that has to be optimized

for a specific operating condition (airspeed and altitude). This geometry is

parameterized using B-splines curve fitting (as described in chapter 2) to obtain

values of the design variables (𝑁𝑣𝑣𝑣 = 1110 control points and the angle of attack).

The range of variation for each of the design variables is defined by assigning an

upper and a lower limit. These limits, which are based on certain structural constraints

(as described earlier), become the design space for the optimization problem.

68

The design space (design variables) is then discretized into a number of levels

(𝑁𝑠) which is equal to the number of CFD simulations to be performed. The 𝑁𝑠

values of each of the design variables are then arranged in a set, 𝐗𝐷𝐷𝐷 as per the LHS

requirements.

CFD simulation is performed for each of the samples in the DoE for a specific

operating condition (airspeed and altitude). The results (𝐶𝐿 ,𝐶𝐷 ,𝐶𝑀,𝐶𝑃𝑥𝑠) are recorded

and using these values, the objective function is evaluated. This process is carried out

for all the samples and a results array, [𝐲], is formed wherein the objective function

values are arranged corresponding to their respective samples as in the DoE.

An over-determined problem is formulated based on the DoE and the

responses (objective function values)— [𝚽][𝛃] = [𝐲] ([𝚽] is the Vandermonde

matrix in terms of the design variables 𝐱 ). This equation is then solved for [𝛃], which

is a matrix consisting of the polynomial regression coefficients, in a linear least-

squares sense. These values of [𝛃] are then used to construct the RSM, 𝐲 ≡ 𝑓(𝐱,𝜷),

which is a quadratic polynomial with 𝑛𝑡 = 78 terms.

The RSM is then optimized using the SQP technique and a set of optimum

values for the design variables, [𝐱𝑶𝑶𝑶], and the corresponding function value, Jappr, is

obtained.

The optimum value needs to be verified, and so the airfoil geometry

corresponding to [𝐱𝑶𝑶𝑶] is analyzed by performing an additional CFD simulation. The

objective function value, using the results of this simulation, is computed Jactual. If the

difference between Jappr and Jactual lies within a certain tolerance value, then [𝐱𝑶𝑶𝑶]

is the minimum.

Else, [𝐱𝑶𝑶𝑶] is added to the initial sampling set 𝐗𝐷𝐷𝐷 and the corresponding

actual function value, Jactual, is added to the results array, [𝐲]. The over-determined

problem [𝚽][𝛃] = [𝐲] is reformulated and solved for [𝛃]. 𝐲 is then re-constructed

with the updated values of [𝛃], and optimized to get a new set of optimum values

[𝐱𝑶𝑶𝑶]. This process of reconstructing the RSM with updated values of [𝛃] is repeated

till the difference between the actual and approximated objective function value does

not within a specified tolerance.

69

Geometry Parameterization • Obtain 𝑁𝑣𝑣𝑣 Design Variables (DV) • B-Spline CPs and Angle of attack.

Design Space Definition • Upper and Lower limits of DV define

the range of variation. • Limits defined using baseline DVs.

Design Space Sampling • Discretizing range of DVs into 𝑁𝑠

levels. • LHS for arranging the 𝑁𝑠 into (𝐗𝐷𝐷𝐷).

Performing Flow Simulation • Meshing and analyzing the flow for

each of the 𝑁𝑠 samples. • Recording the results of the

simulation: 𝐶𝐿 ,𝐶𝐷 ,𝐶𝑀,𝐶𝑃𝑥𝑠

Evaluating Objective Function values

• Evaluating objective function values from obtained results.

• Values arranged in a set based on the DoE, corresponding to each experiment [𝚽][𝛃] = [𝐲].

Developing Initial RSM • Full quadratic polynomial fit 𝐲 ≡ 𝑓(𝐱,𝜷).

• Regression using solution to the Linear Least squares equation.

Optimizing the RSM • Optimize the RSM using

optimization algorithm (SQP).

• Obtain optimum values of DVs [𝐱𝐎𝐎𝐎] and objective value Jappr

Verifying the RSM • Obtain airfoil geometry

from [𝐱𝑶𝑶𝑶] • Perform CFD Simulation—

Get results. • Evaluate Objective Value

Jactual

Check Jappr ≈ Jactual

?

MINIMUM OBTAINED

Add [𝐱𝐎𝐎𝐎] with corresponding Jactual to original DoE

YES

NO

Airflow conditions—Airspeed, altitude (atmospheric pressure)

Initial Airfoil Geometry

Figure 14 Iterative RSO process flow chart

70

3.4 MATLAB Implementation of the Iterative RSO

The Iterative RSO method for the airfoil aerodynamic shape optimization

problem is implemented using MATLAB®, by integrating it with GridPro (meshing

software) and ANSYS® FLUENT (flow solver). This implementation using

MATLAB consists of two main blocks—CFD Block and the OPTIMIZATION

Block.

The CFD Block, as the name implies, consists of all the processes that are

required to perform a single CFD simulation. These include the meshing, flow solving

and results generation processes. For a single CFD simulation, the following steps are

performed via the CFD block—

1. Airfoil coordinate files generation—In this step two files are created that

consist the x, y and z coordinates of the upper and lower surface of the

airfoil (GPro_af1.DAT and GPro_af2.DAT). These files are required for

the mesh generation process.

2. GridPro Topology Input Language (TIL) file—A TIL file is generated that

consists of the commands that need to be executed by GridPro in-order to

generate the mesh (FINAL3.FRA). This file is airfoil geometry specific in

that the topology is constructed around the airfoil geometry (i.e. the

topology varies with different airfoil shapes).

3. Trigger GridPro—MATLAB triggers the GridPro engine (GGrid) that

takes in the coordinate files and the TIL file as the input, performs the

meshing process and returns the mesh file in the required format

(FLO_MESH.msh).

4. ANSYS-FLUENT variable file—Here a variables file is generated that

includes various values pertaining to the operating condition of the flow.

These include the operating pressure (P), ambient temperature (T), Mach

number (M). In addition, the angle of attack in terms of the unit vector

magnitudes in the horizontal and vertical direction of the flow is included

(OP.var).

5. Trigger ANSYS-FLUENT—MATLAB triggers ANSYS-FLUENT which

reads the generated mesh file, loads the variable file, and performs the

71

flow simulation by executing the commands included in the journal file

(FLUENT_RUN3.JOU). This is discussed in the subsequent sections.

6. Result Files—At the end of the simulation, the various results are saved in

individual files. The results include—lift and drag forces, moments,

surface pressure coefficient distribution and surface shear distributions.

(LIFT.CSV, DRAG.CSV, MOMENT.CSV, CP_TOP.CSV,

CP_LOW.CSV, Stress_TOP.CSV and Stress_LOW.CSV).

7. These files are read into MATLAB and the values obtained from these

files are fed into the OPTIMIZATION Block.

The OPTIMIZATION Block, consists of all the processes that are required for

carrying out the various steps in the iterative RSO methodology. These include

geometry parameterization, RSM construction and optimization and the iterative RSO

methodology. The two blocks are interconnected, i.e. data flows to and fro as

required. The following steps are performed via the OPTIMIZATION Block—

1. Airfoil parameterization—In this step, the parameterization of the airfoil is

performed using B-spline curve fitting. Here the control point coordinates

and a curve-parameter set is returned. The y-ordinates of the control points

(along with the angle of attack) are used as design variables (DV) to

perform the optimization process. (optimum_fit.m)

2. DV discretization—A range for the DVs obtained in the above step is

defined in terms of lower and upper limits. The range of each of the DVs is

then discretized into equal number of levels and arranged in a LHS plan

and DoE matrix based on this plan is obtained. (LH_Doe.m, Sampling.m)

3. Objective function evaluation—For each of the planned experiments in the

DoE, the objective function is evaluated. This is done by performing a

CFD simulation for each experiment in the DoE. Here the CFD Block is

used to obtain the values of 𝐶𝐿 ,𝐶𝐷 ,𝐶𝑀 which are used to evaluate the lift-

constrained drag minimization problem. An array of function values is

obtained which correspond to the responses obtained for each of the

experiments.

4. RSM Construction—Based on the DoE and response values (steps 3 and

4), an over-determined problem is formulated [𝚽][𝛃] = [𝐲]. This problem

is then solved for [𝛃], and the RSM, 𝐲 ≡ 𝑓(𝐱,𝜷), is constructed.

72

5. RSM Optimization—Using the MATLAB Optimization and Global

Optimization toolbox function, the RSM is optimized, and the optimum

values of the design variables 𝐱𝑶𝑶𝑶 along with the corresponding objective

function value Jappr(which is the function value approximated by the

RSM) is obtained. Here the SQP algorithm is employed using MATLAB’s

fmincon and MultiStart functions. (Mult_start_opt.m)

6. Model verification—For the verification of the model, an additional CFD

simulation is performed based on the values of 𝐱𝑶𝑶𝑶. Here the airfoil

geometry corresponding to the values of 𝐱𝑶𝑶𝑶 is obtained, and is sent to

the CFD Block where the simulation is performed. The results obtained

from this run are used to evaluate the actual objective function Jactual. This

value is then compared with the approximated value Jappr.

7. If the difference between Jappr and Jactual is within a tolerance limit, then

𝐱𝑶𝑶𝑶 is the final optimum and the corresponding airfoil shape is the

optimized design. Else, the iterative RSO methodology is implemented.

Here 𝐱𝑶𝑶𝑶 and the corresponding Jactual is added to the initial DoE and

the RSM is reconstructed and optimized. This process is repeated till the

difference between Jappr and Jactual does not lie within the tolerance limit.

Here the tolerance is set at 5 percent of Jactual.

The entire process is managed by a main directory called the MATLAB

MASTER, which includes all the M-Files that are required to carry out the airfoil

optimization process. Some of the important M-Files include—

- bsp_basic.m —Code for the generating a B-spline curve given the

coordinates of the control points, parameter values and the order of the

curve. This code returns the x and y coordinates of the B-spline curve.

- optimum_fit.m—Code for the parameterization of the baseline airfoil.

Given the original coordinates of the airfoil geometry and a curve order,

this code returns the coordinates of the control points and parameter values

of the B-spline curve that best fits the baseline airfoil geometry. The y-

ordinates of the control points and the angle of attack, the constitute the

vector of design variables 𝐱.

73

Figure 15 MATLAB implementation of the Iterative RSO method

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74

- CL_req.m—Code for calculating the required coefficient of lift (CL)

given the weight, airspeed (Mach number) and altitude (in meters).

- LH_doe.m—Code for generating a LHS DoE plan required for

constructing the RSM. Given the number of levels required, 𝑁s, and the

upper and lower limits of each of the design variables, the code returns a

DoE set, 𝑿DoE consisting values of the design variables at all levels as per

a LHS design.

- Gramian.m—Code for creating the Vandermonde matrix [𝚽] and

solving for 𝛃, given the values of the design variables at all levels 𝑿DoE

and the corresponding responses [y].

- J_approx.m—Code for evaluating the RSM, 𝐲, given the values of 𝛃

and design variables 𝐱. This returns an approximate value of the objective

function Jappr.

- IterativeRSO.m—Code for implementing the RSO based on the

iterative methodology described above. Given 𝑿DoE , 𝛃 and [y], the code

performs the iterative RSO and returns the optimum values of the design

variables 𝐱𝑶𝑶𝑶.

- NonlinCon.m—Code for evaluating the non-linear constraints that are

imposed on the RSM during optimization.

- Multi_start_opt.m—Code for performing global optimization using

the Multi Start optimization algorithm. Here the optimization is performed

from multiple starting points of the RSM leading to the optimum values.

MATLAB Optimization Toolbox 3.4.1

MATLAB Optimization Toolbox provides widely used algorithms for

standard and large-scale optimization. These algorithms solve constrained and

unconstrained continuous and discrete problems. The toolbox includes functions for

linear programming, quadratic programming, binary integer programming, nonlinear

optimization, nonlinear least squares, systems of nonlinear equations, and

multiobjective optimization [39].

The function used here is fmincon. The purpose of this function is to find the

minimum of a constrained nonlinear multivariable functions. It solves for a minimum

of a problem specified by

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min𝑥𝑓(𝑥) such that

⎩⎪⎨

⎪⎧

𝑐(𝑥) ≤ 0𝑐𝑒𝑞(𝑥) = 0𝐴 ∙ 𝑥 ≤ 𝑏

𝐴𝑒𝑞 ∙ 𝑥 = 𝑏𝑒𝑞𝑙𝑏 ≤ 𝑥 ≤ 𝑢𝑏

Here x, b, beq, lb, and ub are vectors, A and Aeq are matrices, c(x) and ceq(x)

are functions that return vectors, and f(x) is a function that returns a scalar. f(x), c(x),

and ceq(x) can be nonlinear functions.

fmincon attempts to find a constrained minimum of a scalar function of several

variables starting at an initial estimate. This is generally referred to as constrained

nonlinear optimization or nonlinear programming.

The fmincon function syntax is as follows:

[x, fval] = fmincon(FUNC, x0, A, b, Aeq, beq, lb… ,ub,

NONLCON, OPTIONS)

This function call starts the optimization process from an initial guess value x0

and attempts to find a minimizer x of the function described in func subject to the

linear inequalities A*x≤b, equalities Aeq*x≤beq. It also defines a set of lower and

upper bounds on the design variables in x, so that the solution is always in the range

𝑙𝑏 ≤ 𝑥 ≤ 𝑢𝑏. It subjects the minimization to the nonlinear inequalities c(x) or

equalities ceq(x) defined in a function NONLCON.

Certain options required for the optimization such as the algorithm to be

employed etc. is set using OPTIONS, which selects from a structure OPTIMSET. At

the end of the optimization process, the function returns the value of the minimized x

and the value of the corresponding objective value fval.

FUNC, is the objective function to be minimized. FUNC is a function that

accepts a vector x and returns a scalar f, the objective function evaluated at x. FUNC

can be specified as a function handle for a file:

[x] = fmincon(@myfunc,x0,A,b)

where myfunc is a MATLAB function such as

function f = myfunc(x)

f = ... % Compute function value at x

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NONLCON is a function call that computes the nonlinear inequality

constraints 𝑐(𝑥) ≤ 0 and the nonlinear equality constraints 𝑐𝑒𝑞(𝑥) = 0. NONLCON

accepts a vector x and returns the two vectors c and ceq. c is a vector that contains

the nonlinear inequalities evaluated at x, and ceq is a vector that contains the

nonlinear equalities evaluated at x. NONLCON should be specified as a function handle

to a file or to an anonymous function, such as mycon:

x = fmincon(@myfunc,x0,A,b,Aeq,beq,lb,ub,@mycon)

Here mycon is a MATLAB function such as

function [c, ceq] = mycon(x)

c = ... % Compute nonlinear inequalities at x.

ceq = ... % Compute nonlinear equalities at x.

OPTIONS, selects certain optimization parameters from a structure called

OPTIMSET. Some of these options include—

- Algorithm—For selecting the optimization algorithm such as SQP etc.

- DerivativeCheck—For comparing the values of user-supplied

derivatives to finite-differencing derivatives.

- GradObj—For using the gradient for the objective supplied by the user.

- MaxFunEvals—For specifying the maximum number of objective

function evaluations.

- MaxIter—For specifying the maximum number of iterations allowed.

- TolFun—For specifying the termination tolerance on the function value.

The function call employed here is as follows:

f=@(x)rsm_opt(B, x );

nonlcon=@(x)NonlinCon(B,x,J);

[x_opt, fval]=fmincon(f, x0,[],[],[],[], lb, ub, nonlcon…

,options);

Here f is a parameter that calls the function rsm_opt(B, x ) which takes

in values of the design variables x and the regression coefficients 𝛃, and evaluates the

RSM equation. nonlcon is a parameter that calls the function

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NonlinCon(B,x,J) that takes in values of x, 𝛃, and the actual value of the

objective function at the previous step J, and evaluates the non-linear constraints

functions.

The fmincon function call starts the optimization process at x0(initial values

of the design variables to initiate the optimization process), and performs the

optimization iterations such that the iterate at each step lies within the bounds lb and

ub, and satisfies the nonlinear constraints nonlcon. Following are the options

that are used in the function call—

options=optimset('GradObj','on','GradConstr','on',’algori

thm','sqp',MaxFunEvals',10000000,'MaxIter',4000,'TolCon',

… 0.00001,'TolX',0.00001);

The above options imply that the gradient of the objective function and the

constraints are user-supplied, the algorithm to be used is SQP, maximum number of

function evaluations are set at 10000000, maximum number of iterations are set at

4000, the constraints tolerance is set at 0.00001 and the tolerance for the change in the

values of the design variables is set at 0.00001.

MATLAB Global Optimization Toolbox 3.4.2

Global Optimization Toolbox provides methods that search for global

solutions to problems that contain multiple maxima or minima. It includes Global

Search, Multistart, Pattern Search, Genetic Algorithm, and Simulated Annealing

solvers [40].

These solvers can be used to solve optimization problems where the objective

or constraint function is continuous, discontinuous, and stochastic, does not possess

derivatives, or includes simulations or black-box functions with undefined values for

some parameter settings.

Generally, Optimization Toolbox solvers find a local optimum. (This local

optimum can be a global optimum). They find the optimum in the basinof attraction

of the starting point. In contrast, Global Optimization Toolbox solvers are designed to

search through more than one basin of attraction.

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The GlobalSearch and MultiStart solvers apply to problems with smooth

objective and constraint functions. The solvers search for a global minimum, or for a

set of local minima. GlobalSearch and MultiStart work by starting a local solver, such

as fmincon, from a variety of start points. In this case the MultiStart global

optimization function is employed. Following are the functions used to perform

optimization using the MultiStart function—

- CustomStartPointSet.m—This is a set of starting points (starting

values of the design variables x) that is supplied the MultiStart solver. By

doing so, the MultiStart Solver starts a local optimization process from

each of the supplied starting points. These points could be random points

or user-supplied.

- createOptimProblem.m—This function call is used to setup the

MultiStart problem. This is required in order to specify the local

optimization solver to employed, objective functions, constraints, bounds,

options etc.

- run.m—This function call runs the MultiStart solver based on the

problem structure and the starting points defined above.

The functions used in this implementation are as follows:

tpoints = CustomStartPointSet([X_pnts]);

problem = createOptimProblem('fmincon','x0',x0_2,...

'objective',@(x)rsm_opt(B, x ),'lb',lb,'ub',ub,...

'nonlcon',@(x)NonlinCon(B,x,J),'options',options);

ms=MultiStart;

[xmin, J_min]=run(ms,problem,tpoints);

Here tpoints is the set consisting of the starting points defined by the

CustomStartPointSet function, using the supplied points X_pnts.

problem is the optimization structure defined by the

createOptimProblem function. It implies that the local optimization solver to be

employed is the fmincon solver, the objective function is defined by the function

rsm_opt(), the lower and upper bounds are defined by lb and ub, the nonlinear

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constraints are defined by nonlcon, and the local minimization options are defined

by options. Here fmincon, rsm_opt(), nonlcon are just as defined in

the previous sub-section.

ms sets the global optimization solver to MultiStart. run performs the

optimization process using ms, problem and tpoints that are defined above. At the end

of the process, the lowest value of the objective function, J_min (which is the least

amongst all the minima), and the corresponding values of the design variables xmin

are returned.

3.5 Meshing using GridPro

For performing an external flow simulation using the CFD technique, the

region around the airfoil to be meshed (spatial discretization) into quadratic cells.

Here an automatic grid generator—GridPro is used to generate the airfoil meshes

which are then used to carry out the flow simulation using ANSYS-Fluent.

For a multi-block structured grid generator, automation can be classified into

four areas: 1). optimum distribution of high quality grid, 2). book keeping of

topological information, 3). topology generation, and 4). surface restructuring and

repair.

To this end, GridPro is a general purpose, 3-dimensional, multi-block

structured grid (mesh) generator using an advanced smoothing scheme that

incorporates many automatic features.

To generate a mesh, the input required from a user has three components—

surface specifications, a block topology and a run schedule. These are described as

follows:

Surface Specifications 3.5.1

The surface specifications constitute an external component in that they are

provided mostly from the outside of the GridPro environment and conform to certain

standards. The surface is basically the geometry around which the meshing is to be

done and they are independent of the surface grid generated within GridPro. These

surfaces are basically data files (file extension .dat) that contain the x, y and z

coordinates surface points. The first line of the file displays the number of points used

to describe the surface followed by the x, y and z coordinates of the points in three

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columns. These files are then imported into the GridPro topology engine via the

Topology Input Language (TIL) file.

For meshing the airfoil a C-type grid is to be generated. The C-type refers to

the C-shape of the far-field boundary. The radius of this boundary is taken as 20 times

the chord-length of the airfoil. The following surfaces were used to build grid—

- A C-shaped surface (dark blue) to define the boundary of the grid

(filename: C_L.dat). This surface is assigned the pressure far-field

property boundary condition so as to be recognized accordingly by the

Flow-solver (ANSYS FLUENT).

- A rear-end surface (purple) so as to limit the boundary at the rear of the

airfoil. This surface is also assigned the pressure far-field boundary

condition property (filename: Back_end.dat).

Figure 16 GridPro surfaces

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- The airfoil surface (light green) is split into two—an upper surface and a

lower surface, with a coordinates data file for each of them (GPro_af1.dat

and GPro_af2.dat). These surfaces are assigned the wall property so that

the flow solver applies the appropriate wall boundary conditions to them.

- In addition to the above surfaces, certain internal surfaces are specified.

An internal surface is a surface for which both sides of the surface are to

be gridded. These surfaces are used to provide the required grid point

distribution. If a region within the grid the is over stretched or if the

geometry contains great disparity in length scales and curvatures, internal

surfaces are used to separate the regions of disparity and get better grid

distribution. Here, it is also used to capture the features of the airfoil

wakeline (aft of the airfoil). This is done by creating a surface which starts

from the airfoil trailing edge, and extends through to the rear end surface.

Also, this surface must be created such that it bisects the airfoil trailing

edge angle. Using this internal surface also ensures a sharp trailing edge

for the airfoil (filenames: af_lead.dat, trail_top.dat, trail_down.dat,

Wake_line.dat).

Block Topology 3.5.2

After the surfaces have been defined, the next task is to design a block

topology for the region to be gridded. This can be done with the GridPro/az-Graphic

Manager, which is the graphical user interface for the GridPro engine.

The goal is to cover the region with quadrilaterals (blocks). This covering does

not need to be done at a geometric precise level. It needs to be done only at a rather

topological level. That is, a surface can be represented as a set of piece-wise linear

segments placed not too far from the real surface.

It can be noticed that the entire region between the outer far-field and the

airfoil at the center has been filled with blocks (quadrilaterals). Here the points in

orange are referred to as corners and the yellow line segments as block edges. Each

corner in the interior of the domain is shared by four neighboring blocks and each

block edge is shared by two blocks. The corners and edges are arranged such that

quadrilateral that are eventually formed capture the topological features (geometry) of

the surfaces.

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The topology building is started from the outer far-field by placing the corners

(numbered 0, 128, 115 and so on) around in close proximity to the outer surfaces.

These corners are then linked to form edges. Corners (numbered 127, 114, 101 and so

on) are added in the interior of the domain in the manner shown in the above figure.

These corners are then connected to each other and to the corners surrounding the

outer surfaces such that blocks (quadrilaterals) are formed. More corners and edges

are placed in the interior of the domain closing in on to the center such that blocks

now begin to wrap around the airfoil surface and the wake-line.

The region surrounding the airfoil and the wake-line is where the flow is

highly dynamic and turbulent. So it is important to capture the geometrical features in

these regions with greater accuracy and thus more blocks are used to wrap the airfoil.

Also higher grid densities (number of cells emerging from the edge) are assigned to

edges that make up these blocks. This leads to a relatively denser grid near the airfoil

surface and the wake line.

Figure 17 Topology layout for the airfoil and the far-field

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Figure 18 Topology layout in the wake region of the airfoil

It is desirable to have the blocks as orthogonal as possible in the regions

surrounding the airfoil so that the resulting grid around it is also orthogonal. This

leads to more accurate CFD results. In order to do so, a MATLAB code is written

which adjusts the position of the corners surrounding the airfoil and the wake-line.

The corners immediately surrounding the airfoil are positioned such that the edges

emerging from them (radially) are perpendicular to the airfoil surface. Also, the

arrangement of these corners ensures that the edges connecting them are

approximately tangential to the surface. This way, the blocks are almost tangential to

the airfoil surface and thus the cells that are formed are almost orthogonal to the

surface.

Once the topology design and layout is complete, the corners in the

immediate vicinity of the surface are assigned to it. For instance, the corners that are

closest to the airfoil surface are assigned to it. This way the GridPro grid engine

recognizes the points which are associated to surfaces and wraps the grid around them

accordingly.

Figure 19 Topology around the airfoil surface

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After the completion of the topology creation and the assignment of the

respective surface properties, the topology is saved as a Topology Input Language

(TIL) file (extension .FRA). This file keeps a record of the coordinates of all the

corners, edges, surface filenames, properties, grid densities etc. and is used to run the

gridding process. (Filename: FINAL3.FRA).

Run schedule 3.5.3

Since GridPro uses an advanced smoothing scheme in which the grid is

generated in multiple sweeps just as in the case of the ordinary elliptic grid

generation, a schedule is provided for a better and faster convergence. Here 1000

sweeps are used to generate the grid.

A schedule file consists of step lines, each of which lists a sequence of actions

that direct the run process of GridPro. The name of a schedule file has the prefix part

same as the corresponding main topology file and end with the file name extension

'.sch' (Filename: FINAL3.sch).

The GGrid engine is run which executes the TIL file and performs the

specified number of sweeps. At the end of the sweeping process a temporary block

file is generated (extension .tmp). This file includes the coordinates of the grid points

that are generated at the last sweep.

Wall Clustering 3.5.4

After the grid is generated (after executing the TIL file with the sweeps), wall

clustering is used to adjust the spacing of the cells near the airfoil surface such that

there is a much denser grid near the surface. This is particularly important in order to

capture the flow physics associated with turbulent and separated flows.

Wall clustering reduces the normal spacing of the cells attached to the wall

surfaces to a much lower value. It then starts increasing the spacing of the cells in the

layers adjacent to the wall layer by a stretch factor called the growth raito. This way

there is a gradual increment in the cell spacing starting from the wall surface going

towards the outer far field. This is done using the –clu command. Here the spacing of

the wall cells (cells attached to the wall surface) is set at 0.00025 and the growth ratio

is set at 1.05.

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After the clustering is performed, the temporary grid is now saved in the

required format which can be read by the flow solver. The ANSYS FLUENT has a

specific format for the grid which is given by the (.MSH) extension. A FLUENT

output script (outU_fluent.script.bat) is run inside GridPro which converts the

temprorary (.tmp) grid file into the required (.MSH) format. This mesh can then be

imported into ANSYS FLUENT to perform the required simulation.

Figure 20 Mesh—full display

Figure 21 Mesh—full and close up view

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3.6 CFD Solver—ANSYS FLUENT

ANSYS FLUENT is a flow solver that provides comprehensive modeling

capabilities for a wide range of incompressible and compressible, laminar and

turbulent fluid flow problems. Steady-state or transient analyses can be performed. In

ANSYS FLUENT, a broad range of mathematical models for transport phenomena

(like heat transfer and chemical reactions) is combined with ability to model complex

geometries [31].

Robust and accurate turbulence models are a vital component of the ANSYS

FLUENT suite of models. The turbulence models provided have a broad range of

applicability, and they include the effects of other physical phenomena, such as

buoyancy and compressibility.

For all flows, ANSYS FLUENT solves conservation equations for mass and

momentum which are the basic governing equations for any flow. For flows involving

heat transfer or compressibility, an additional equation for energy conservation is

solved. In addition to these, turbulence model equations are solved to account for the

turbulence associated with aerodynamic flows. The turbulence model used here is the

Spalart-Allmaras model. It is a one-equation turbulence model and it is solved for the

turbulent kinematic viscosity 𝜈, which can then be used to calculate the turbulent

dynamic viscosity, 𝜇𝑡, that will eventually close the RANS equations. All these

equations have been discussed in depth in Chapter 2.

In ANSYS FLUENT, the flow can be solved using one of the two numerical

methods— Pressure-based solver and the Density-based solver. In either of the cases

a control-volume based technique is used that consists of—

- Division of the domain into discrete control volumes using a

computational grid.

- Integration of the governing equations on the individual control volumes to

construct algebraic equations for the discrete dependent variables

(unknowns) such as velocities, pressure, temperature, and conserved

scalars.

- Linearization of the discretized equations and solutions of the resultant

linear equation system to yield updated values of the dependent variables.

87

Pressure-Based Solver 3.6.1

In the pressure-based approach, the pressure field is extracted by solving a

pressure or pressure correction equation which is obtained by manipulating the

continuity and momentum equations. Each iteration of the solver consists of the

following steps

- Update fluid properties (eg. density, viscosity, turbulent viscosity etc.)

- Solve simultaneously the coupled system of equations—momentum and

pressure based continuity equations.

- Correct face mass fluxes, pressure and the velocity field.

- Solve the equations for additional scalars, such as turbulent quantities etc,

- Check for convergence of the equations.

These steps are continued until the convergence criteria are met.

General Scalar Transport Equation 3.6.2

All the governing differential equations for fluid motion, which are based on

the conservation principles (conservation of mass, momentum and energy) can be

written in a generic form. Upon inspection of the all the conservation equations, it can

be inferred that all the dependent variables seem to obey a generalized conservation

principle. If the dependent variable (scalar) is denoted by 𝜙, the generic differential

equation is given as

𝜕𝜌𝜙𝜕𝑡

𝑇𝑣𝑣𝑛𝑠𝑖𝑒𝑛𝑡 𝑡𝑒𝑣𝑚

+ 𝛁 ∙ (𝜌𝐕𝜙)𝐶𝐷𝑛𝑣𝑒𝑐𝑡𝑖𝐷𝑛 𝑇𝑒𝑣𝑚

= 𝛁 ∙ (Γ𝛁𝜙)𝐷𝑖𝑓𝑓𝑢𝑠𝑖𝐷𝑛 𝑡𝑒𝑣𝑚

+ 𝑆𝜙𝑆𝐷𝑢𝑣𝑐𝑒 𝑡𝑒𝑣𝑚

Here Γ is the diffusion coefficient or diffusivity.

- The transient term, 𝜕𝜌𝜙𝜕𝑡

, accounts for the accumulation of 𝜙 in the

concerned control volume.

- The convection term, 𝛁 ∙ (𝜌𝐕𝜙), accounts for the transport of 𝜙 due to the

existence of the velocity field.

- The diffusion term, 𝛁 ∙ (Γ𝛁𝜙), accounts for the transport of 𝜙 due to its

gradients.

- The source term, 𝑆𝜙, accounts for any sources or sinks that either create or

destroy 𝜙. Any extra term that cannot be cast into the convection or

88

diffusion terms are considered as source terms. For chemically inert flows,

such as aerodynamic flows, where ideal gases are assumed in the

calculations, the 𝑆𝜙 term is equated to zero.

The objective of all discretization techniques (here the finite volume method)

is to devise a mathematical formulation to transform each of these terms into an

algebraic equation. Once applied to all control volumes in a given mesh, a full linear

system of equations is obtained that needs to be solved for the variable 𝜙.

ANSYS FLUENT uses a control-volume based technique to convert a general

scalar transport equation to an algebraic equation can be solved numerically. This

control volume technique consists of integrating the transport equation about each

control volume, yielding a discrete equation that expresses the conservation law on a

control-volume basis.

This is demonstrated by the following equation (which is the general transport

equation) written in integral form for an arbitrary control volume V as follows:

𝜕𝜌𝜙𝜕𝑡

𝑑𝑉𝑉

+ 𝜌𝜙𝐕 ∙ 𝐝𝐀 = Γ𝜙𝛁𝜙 ∙ 𝐝𝐀 + S𝜙𝑑𝑉𝑉

Where 𝜌 = density

𝐕 = velocity vector (𝐕 = 𝑢𝐢 = 𝑣𝐣)

𝐝𝐀 = differential surface area vector

Γ𝜙 = diffusion coefficient for 𝜙

∇𝜙 = gradient of 𝜙 = (𝜕𝜙 𝜕𝑥⁄ )𝐢 + (𝜕𝜙 𝜕𝑦⁄ )𝐣

S𝜙 =source for 𝜙 per unit volume

The above equation is applied to each control volume, or cell, in the

computational domain. Discretization of the above equation on a given cell yields

𝜕𝜌𝜙𝜕𝑡

𝑉 + 𝜌𝑓𝐕𝑓𝜙𝑓 ∙ 𝐀𝑓

𝑁𝑓𝑎𝑐𝑒𝑠

𝑓

= Γ𝜙𝛁𝜙𝑓 ∙ 𝐀𝑓

𝑁𝑓𝑎𝑐𝑒𝑠

𝑓

+ S𝜙𝑉

Here

- 𝑁𝑓𝑣𝑐𝑒𝑠 = number of faces enclosing cell (for a quad cell 𝑁𝑓𝑣𝑐𝑒𝑠 = 4).

89

- 𝜙𝑓 = value of 𝜙 convected through face f.

- ∑ 𝜌𝑓𝐕𝑓𝜙𝑓 ∙ 𝐀𝑓𝑁𝑓𝑎𝑐𝑒𝑠𝑓 = mass flux through the face.

- 𝐀𝑓 = area vector of the face (𝐀𝑓 = 𝐴𝑥𝐢 + 𝐴𝑦𝐣)

- 𝛁𝜙𝑓 = gradient of 𝜙 at face f.

- 𝑉 = cell volume.

The discretized scalar transport equation contains the unknowns scalar

variable 𝜙 at the cell center as well as the unknown values in surrounding neighbor

cells. This equation will, in general, be non-linear with respect to these variables. A

linearized form can be written as

𝑎𝑃𝜙 = 𝑎𝑛𝑏𝜙𝑛𝑏𝑛𝑏

+ 𝑏

Here the subscript nb refers to neighbor cells, and 𝑎𝑃 and 𝑎𝑛𝑏 are the

linearized coefficients for 𝜙 and 𝜙𝑛𝑏. The number of neighbors for each cell depends

on the mesh topology, but will typically equal the number of faces enclosing the cell

(with the exception of the boundary cells).

Similar equations can be written for each cell in the mesh. This results in a set

of algebraic equations with a sparse co-efficient matrix. For scalar equations, ANSYS

FLUENT solves the linear system using a point implicit (Gauss-Siedel) linear

equation solver in conjunction with an algebraic multigrid (AMG) method.

Spatial Discretization 3.6.3

By default, ANSYS FLUENT stores discrete values of the scalar 𝜙 at the cell

centers. However, face values 𝜙𝑓 are required for the convection terms and must be

interpolated from the cell center values. This is accomplished using an upwind

scheme.

Upwinding means that the face value 𝜙𝑓 is derived from quantities in the cell

upstream, relative to the direction of the normal velocity 𝐕𝑓. There are several upwind

schemes that are available—first-order upwind, second-order upwind, power law and

QUICK. Here the second-order upwind scheme is used.

90

In this scheme, the quantities at the cell faces are computed using a

multidimensional reconstruction approach. In this approach, higher-order accuracy is

achieved at cell faces through a Taylor series expansion of the cell-centered solution

about the centroid. Thus the face value 𝜙𝑓 is computed using the following

expression—

𝜙𝑓 = 𝜙 + 𝛁𝜙 ∙ 𝐫

Here 𝜙 and 𝛁𝜙 are the cell-centered value and its gradient in the upstream

cell, and 𝐫 is the displacement vector from the upstream cell centroid to the face

centroid. This formulation required the determination of the gradient 𝛁𝜙 in each cell.

Computing Forces and Moments 3.6.4

For the wall surfaces, the forces along a specified vector and the moments

about a specified center and axis are computed.

The total force component along a specified force vector a on a wall surface

(airfoil surface) is computed by summing the dot product of the pressure and viscous

forces on each surface with the specified force vector. The terms in this summation

represent the pressure and viscous force component in the direction a:

𝐹𝑣⏟𝑇𝐷𝑡𝑣𝑙 𝑓𝐷𝑣𝑐𝑒 𝑐𝐷𝑚𝑝𝐷𝑛𝑒𝑛𝑡

= 𝒂 ∙ 𝐅𝑝𝑝𝑣𝑒𝑠𝑠𝑢𝑣𝑒 𝑓𝐷𝑣𝑐𝑒 𝑐𝐷𝑚𝑝𝐷𝑛𝑒𝑛𝑡

+ 𝒂 ∙ 𝐅𝑣𝑣𝑖𝑠𝑐𝐷𝑢𝑠 𝑓𝐷𝑣𝑐𝑒 𝑐𝐷𝑚𝑝𝐷𝑛𝑒𝑛𝑡

Here 𝒂 = specified force vector

𝐅𝑝 = pressure force vector

𝐅𝑣 =viscous force vector

The total moment vector about a specified center A is computed by summing

the cross products of the pressure and viscous force vectors for each surface with the

moment vector 𝐫𝐴𝐵, which is the vector from the specified moment center A to the

force origin B. The terms in this summation represent the pressure and viscous

moment vectors.

𝐌𝐴𝑇𝐷𝑡𝑣𝑙 𝑚𝐷𝑚𝑒𝑛𝑡

= 𝐫𝐴𝐵 × 𝐅𝑝𝑝𝑣𝑒𝑠𝑠𝑢𝑣𝑒 𝑚𝐷𝑚𝑒𝑛𝑡

+ 𝐫𝐴𝐵 × 𝐅𝑣𝑣𝑖𝑠𝑐𝐷𝑢𝑠 𝑚𝐷𝑚𝑒𝑛𝑡

Here

𝐴 = specified moment center

91

𝐵 = force origin

𝐫𝐴𝐵 = moment vector

In addition to the forces and the moments, the associated force and moment

coefficients are also computed for each of the wall surfaces, using the reference

values.

The force coefficient is defined as the force 𝐹𝑣 divided by 12𝜌𝑣2𝐴, and the

moment coefficient is defined as magnitude of the moment 𝑀𝐴 divided by 12𝜌𝑣2𝐴𝐿.

Here 𝜌, 𝑣,𝐴, and 𝐿 are the density, velocity, area the length of the surface.

For calculating the lift and drag on the airfoil, the vector 𝒂 is the unit vector

perpendicular and along the direction of the airflow respectively. For calculating the

moment, 𝐫𝐴𝐵 is the distance vector between the line of action of force and the

reference point, taken to be 0.25L.

ANSYS FLUENT Setup and Run 3.6.5

To perform each CFD run, certain steps need to be taken. These are as

follows—

1. Import the mesh file into the solver—Here the mesh file that is generated

using GridPro is imported into the flow solver using the Read Mesh

command.

2. Define viscous model—Here the turbulence model (Spalart-Allmaras)

with the default values is selected.

3. Define fluid material—Here the type of fluid properties are selected (ideal

gas, air). For defining the viscosity, the Sutherland (3-Co-efficient) method

is used. The ideal gas property automatically activates the Energy

Equation option.

4. Merge-zones—Certain zones in the mesh are merged into one, so that the

total number of zones are reduced. In particular, the pressure far-field and

the interior zones are merged.

5. Load variable file—The variable file is loaded into the solver. This

contains variable values for the operating pressure (P), Temperature (T),

Mach number (M), unit vector magnitudes for the x and y components of

the airflow (M_x and M_y), unit vector magnitudes for the x and y

92

components of the Lift force (L_x and L_y), and unit vector magnitudes

for the x and y components for the Drag force (D_x and D_y).

6. Define boundary conditions—Here the boundary conditions are defined

using the variables that are loaded into the solver (step 5). Here Mach

number and its x and y components, and temperature are defined by

assigning the corresponding variables to the respective fields. Also the

turbulence specification method is set to Turbulent-Viscosity ratio and the

value is set at 10. The operating pressure is defined by assigning the

corresponding variable.

7. Set Solution methods—here the pressure-velocity coupling is set to

coupled. For the Spatial Discretization, the gradient is computed using

Least-Squares Cell Based scheme, Pressure is solved using the standard

scheme, and density, momentum, modified turbulent viscosity and energy

are solved using the second-order upwind scheme.

8. Set solution controls—The Courant Number is set at 200, the explicit

relaxation factors for momentum and pressure are set at 0.5, and under-

relaxation factor for density is set at 0.5, for body-forces, turbulent-

viscosity and energy it is set at 1, and for modified turbulent viscosity it is

set at 0.9.

9. Set monitors—Here the monitors for the continuity, x-velocity, y-velocity,

energy and kinematic viscosity are set. The absolutely convergence criteria

for these variables are respectively set at 1e-07. 1e-05, 1e-05, 1e-06 and

1e-05. In addition to these monitors, the lift, drag and moment residuals

are also monitored.

10. Initialize—The flow field must be initialized in order to carry out the

iterations. This is done by using the hybrid initialization option.

11. Solve—Set the number of iterations and run the solver. Here the number

of iterations is set at 300.

12. Report forces—Once the solution has converged, the wall forces are

reported and saved. The wall forces are the lift, drag and moments. For the

calculation of the correct lift and drag values, the horizontal and vertical

component unit vector magnitudes are specified by the variables L_x and

L_y, D_x and D_y respectively. These variables take their values from the

variable file which is loaded into the solver at the beginning of the

93

program. The moments are calculated about an axis perpendicular X-Y

plane and at 25% of the chord length (LIFT.csv, DRAG.csv, and

MOMENT.csv).

13. The pressure coefficient (CP), Mach number and shear distributions over

the top and bottom surfaces of the airfoil are plotted as a function of

distance along the airfoil chord. These distributions are also saved in

individual files (CP_TOP.csv, CP_LOW.csv, Mach_TOP.csv,

Mach_LOW.csv, Stress_TOP.csv, Stress_LOW.csv).

14. After saving the results, the solver is exited.

The steps mentioned above are recorded in a script file, which is referred to as

a Journal file (extension .JOU). This file is read by ANSYS-FLUENT solver and the

steps are executed automatically without requiring any input from the user. In this

way complete automation is achieved which is the backbone of the parametric

optimization process.

Flow Solver Validation 3.6.6

Flow results verification studies are performed in order to verify the results

obtained from the flow solver. Here the results obtained from ANSYS-FLUENT are

compared with those obtained by verified sources that are usually used by the CFD

community. These resources have been published by NASA’s National Program for

Applications-Oriented Research in CFD (NPARC). NPARC provides an archive of

examples, verification and validation cases for general purpose CFD usages.

The case used for verification here is a RAE 2822 airfoil at transonic

conditions. This case is Study#4 from the NPARC Validation Archive. The

freestream conditions for this case are:

- Mach Number = 0.729 [-]

- Static Pressure = 15.8073 [psi] = 108987.768 [N/m2]

- Temperature = 460.0 [R] = 255.55 [K]

- Angle of attack = 2.31o

A CFD mesh is obtained for the RAE 2822 airfoil using GridPro. The process

is as described in the previous section. The mesh obtained is the analyzed for the flow

based on the steps described above and using the freestream conditions for this case-

94

study. The results obtained are compared with three results published by NPARC.

These are—the NPARC solver code, WIND-US 3.0 code and published wind-tunnel

data. Here, the pressure coefficient (CP) distribution over the upper and lower surface

obtained by FLUENT are compared with the published results as below:

It can be seen from the above plot that the results obtained by ANSYS-

FLUENT are in close agreement with the published results.

3.7 Simulation Time

The processor used to perform the CFD simulations using ANSYS-FLUENT

and GridPro, and the optimization processes using MATLAB has the following

configuration—

- Intel® Core™ i5 CPU, M430

- Processor speed— 2.27 GHz

- Installed memory (RAM)— 8.00GB of which 7.86 GB is usable.

- System type— 64-bit operating system,

- Manufacturer— DELL

The time required for a generating a single mesh is 81.0847 seconds computer

time, while the time required for performing a single CFD flow simulation on the

mesh is 258.037225 seconds computer time. Therefore one complete CFD simulation

(meshing and flow solving) takes 339.1219 seconds of computer time.

Figure 22 CFD verification results

95

The overall meshing and simulation time for 225 samples is 76302.44973

seconds or 21.1951 hours computer time.

96

4. RESULTS AND DISCUSSIONS

4.1 Overview

The iterative RSO methodology is implemented here on various case studies.

First a design validation study is carried out by implementing the scheme for the

optimization of the RAE 2822 airfoil for a particular operating condition. Following

this study, an adaptive airfoil is designed by optimizing a Boeing 737 airfoil for four

different operating conditions. These operating conditions correspond to three pairs of

cruising speeds and altitudes.

The objective function used here is the lift-moment constrained drag

minimization problem. 11 design variables are used—the 10 B-spline control points,

controlling the airfoil shape, and the angle of attack. The initial RSM is constructed

using full quadratic polynomial regression comprising 78 terms and using 200

sampling points based on a LHS design. SQP optimization algorithm is used to

optimize the RSM and the iterative RSO methodology is employed to improve the

predictability of the RSM.

4.2 Case Study#1: Optimization of the RAE 2822 Airfoil

This study is used as a design validation study to compare the results obtained

by the implementation of the iterative RSO methodology with results that have been

published in the literature.

RAE 2822 is a transonic airfoil and it has extensively been used in the

literature as a design validation platform for validating various optimization

methodologies. The design condition of this airfoil is as follows—

- 𝑀∞ = 0.73[−]

- 𝛼 = 2.7°

- 𝑅𝑒 = 7.196 × 106[−]

- Static pressure = 43765 [N/m2]

- Temperature = 300 [K]

For this condition the ANSYS-FLUENT solver has predicted a 𝐶𝐿 value of

0.78902, a 𝐶𝐷 value of 0.0166, and a 𝐶𝑀 value of 0.0918. The viscous, Garabedian

and Korn (VGK) flow-solver code employed by Armando et al. for the same

conditions as above predicted a 𝐶𝐿 value of 0.7894 and a 𝐶𝐷 value of 0.0150 [38]. The

97

Navier-Stokes solver with Baldwin-Lomax turbulence model employed by Ahn et al.

predicted the same value of the lift coefficient (𝐶𝐿=0.7894) and a 𝐶𝐷 value of 0.0193

at 𝛼 = 2.7° [34]. Despite the difference in the drag prediction, all three flow solvers

are in good agreement as far as the pressure coefficient distribution is concerned.

The parameterization of the airfoil, as discussed earlier was carried out using

B-spline control points. The y-ordinates of the control points 2 through 6 and 9

through 13 form the vector of design variables. In addition the angle of attack is taken

as the eleventh design variable.

The range for each DV was obtained by defining an upper and a lower limit.

These were taken as 50% above and below the baseline values respectively.

Figure 23 B-spline parameterization of the RAE 2822 airfoil

Figure 24 Range of the B-spline control points (Design variables)

98

The baseline, and the upper and lower limit values of the design variables are

as follows:

Table 2 Baseline and Upper and Lower limit Values of the Design Variables

DV CP Baseline Values Upper Limit Lower Limit 1 2 0.0161118307 0.02416774612 0.0080559153 2 3 0.0471756500 0.07076347511 0.0235878250 3 4 0.0710297412 0.10654461194 0.0355148706 4 5 0.0633427087 0.09501406311 0.0316713543 5 6 0.0290058511 0.04350877666 0.0145029255 6 9 -0.0304069342 -0.04561040142 -0.0152034671 7 10 -0.0665061285 -0.09975919278 -0.0332530642 8 11 -0.0667191889 -0.10007878335 -0.0333595944 9 12 -0.0101262400 -0.01518936007 -0.0050631200 10 13 0.0038518768 0.00192593840 0.0057778152

11 Angle of Attack

2.70 2 3.4

Following are the freestream conditions for the RAE 2822 design conditions—

Table 3 Freestream conditions for the design operating condition

The following values compare the performance parameters (Lift, moment and

drag) of the baseline and optimized airfoil. Here (C-W) refers to Clockwise moment

and (C-CW) refers to counter-clockwise moments—

True Airspeed Mach 0.73 [-]

253.374 [m/s] Atmospheric Pressure 43765 [N/m2] Air Density 0.508237 [kg/m3] Ambient Temperature 300 [K] (17 [oC])

Dynamic Viscosity 1.7894e-05 [kg/m-s] Reynolds Number 7.2x106 [-]

Angle of Attack 2.7 o

99

Table 4 Baseline and Optimized performance at off-design operating point

PERFORMANCE PARAMETERS BASELINE OPTIMIZED REMARKS

LIFT FORCE Pressure Lift [N] 12872.11 13254.782 Viscous Lift [N] -0.07309 -0.157937 Total Lift [N] 12872.036

[REQUIRED] 13254.624 DIFFERENCE OF

2.97%(HIGHER)

Pressure CL [-] 0.78902 0.812478 Viscous CL [-] -4.4803e-06 -9.681e-06 Total CL [-] 0.789017

[REQUIRED] 0.812469 DIFFERENCE OF

2.97%(HIGHER)

MOMENTS Pressure Moment [N-m]

1499.3075 1675.9315

Viscous Moment [N-m]

-1.1438 -2.05919

Total Moment [N-m]

1498.163 [C-CW] [REQUIRED]

1673.8723 DIFFERENCE OF 11.72%(LESS C-

CW) Pressure CM [-] -0.091903 0.102729 Viscous CM [-] -7.011e-05 -0.00012622 Total CM [ -] 0.091832 [C-CW]

[REQUIRED] 0.1026034 DIFFERENCE OF

11.72%(LESS C-CW)

DRAG FORCE Pressure Drag [N] 180.6682 113.355 Viscous Drag [N] 90.24008 93.63396 Total Drag [N] 270.9082 206.989 REDUCTION OF

23.59%

Pressure CD [-] 0.0110744 0.00694832 Viscous CD [-] 0.0055314 0.0057394 Total CD [-] 0.01660587 0.01268 REDUCTION OF

23.59%

∆ CD = 0.003925 ≈ 4 [DRAG COUNTS] LESS ANGLE OF ATTACK 2.7o 2.24o

100

Table 5 Baseline and Optimized values of the Design Variables

DV CP Baseline Values Optimum Values

1 2 0.0161118307 0.0241677461253

2 3 0.0471756500 0.0425305732457 3 4 0.0710297412 0.0805926624195

4 5 0.0633427087 0.0525445332621 5 6 0.0290058511 0.0354074663741 6 9 -0.0304069342 -0.0177439150251

7 10 -0.0665061285 -0.0595950656119 8 11 -0.0667191889 -0.0446611130085

9 12 -0.0101262400 -0.0110180170713 10 13 0.0038518768 0.0054364685667

11 Angle of Attack 2.70 2.24

Figure 25 RAE 2822 Baseline and Optimized airfoil shapes CP

distributions

101

From the surface CP distributions it can be seen that the adverse pressure

gradient has been significantly reduced, resulting in an almost-shock free airfoil. This

has also resulted in reduced overall drag.

Fitness Parameters

- Adjusted root mean square error σ𝑣

σ𝑣 = 0.467095

This is a small value compared to the data. Thus the model is good enough.

- Coefficient of multiple determination 𝑅𝑣𝑑𝑗2

𝑅𝑣𝑑𝑗2 = 0.94259

This value of 𝑅𝑣𝑑𝑗2 is close to 1. Thus it indicates a good model fit.

4.3 Adaptive Airfoil Design based on the Boeing-737C airfoil

The iterative RSO methodology is employed for the optimization of a Boeing-

737-300 Classic airfoil, taking it as a representative case in an attempt to demonstrate

the advantages of using an adaptive airfoil over a conventional fixed airfoil.

The airfoil is of a near-symmetric shape and is one of 737’s inboard mid-span

airfoil, located between the engine and the aircraft fuselage.

The Maximum Take-Off weight of the aircraft is 62,820[kg] and the total

wing planform area is 105.4 [m2]. It has a cruising speed of Mach 0.74 at 35,000[ft]

(Service ceiling of 37,000 [ft]).

Figure 26 Plot comparing the actual and predicted function values

102

Operational Envelope 4.3.1

The flight envelope or performance envelope of an aircraft refers to the

capabilities of a design in terms of airspeed and load factor or altitude. A doghouse

plot generally shows the relation between speed at level flight and altitude although

other variables are also possible. The plot typically looks something like an upside-

down U and is commonly referred to as a doghouse plot due to its resemblance to a

doghouse. A typical flight envelope for a subsonic (or transonic) aircraft is shown

below:

The outer edges of the diagram, the envelope, show the possible conditions

that the aircraft can reach in straight and level flight. The low (stall) speed boundary

indicates that the aircraft wing can only produce enough lift for a given speed at

various altitudes. The stalling speed (in terms of true airspeed) increases with altitude

(as the density reduces).The service ceiling is the maximum altitude that the aircraft

attains while climbing at a very small climb rate (typically 100 [ft/min]. according to

FAR Part 25 regulations).

In the maximum operating Mach number region, the aircraft reaches a region

of flight where the drag produced increases sharply (i.e., drag divergence Mach

number boundary) and thus the aircraft engines are incapable of producing enough

thrust to accelerate the aircraft to faster speeds. The design of all aircraft structures

carries an assumption about the maximum loads that can be tolerated in flight. This is

limited by the dynamic pressure limit boundary. The bird-strike limit boundary is

Figure 27 Typical aircraft flight envelope

103

Figure 28 Discretized points in the flight envelope.

defined by taking into account the bird-hits on the wind-shields below 10,000[ft].

Traditionally this boundary has been set at 250 knots.

CL requirements across the envelope 4.3.2

The flight envelope is discretized into discrete points, in that various altitude-

Mach number points are obtained within the flight envelope. Given a specific load

requirement [per unit area of the wing], the required values of the lift coefficients are

obtained at these points. This is as shown below—

Here the region bounded by the flight envelope is discretized uniformly and

the various operating points are obtained. However, only those operating points are

chosen for the purpose of optimization, which have a greater probability of being

visited by a typical subsonic transport airplane. The standard cruising point is Mach

0.74 and altitude H= 10670[m]. So it is assumed that the baseline airfoil has been

optimized for this operating point. For the adaptive airfoils problem, three off-design

points are chosen—

A) Mach 0.74 and altitude H = 9083.3 [m]

B) Mach 0.65 and altitude H = 9083.3 [m]

C) Mach 0.65 and altitude H = 6825.0 [m]

The corresponding values for the CL at these points are—A) 0.5305, B)

0.6925, and C) 0.4996. These values are obtained using the lift force obtained from

the airfoil at the standard cruising condition, which is ≈ 6150 [N].

104

Airfoil Geometry 4.3.3

The airfoil geometry for the Boeing airfoil is obtained from the airfoil

database which has been published on the University of Illinois-Urbana Campaign

(UIUC) website. It is a near-symmetric airfoil, with very little rear-loading and strong

shockwaves exist on the upper-surface of the airfoil near the leading edge.

Figure 29 Boeing 737 airfoil (baseline)

The parameterization for this airfoil is performed using B-spline curve fitting

consisting of 10 control points (5 for each surface). This is as shown below

The range for each DV, as before was obtained by defining an upper and a

lower limit. However, in this case the upper and lower limits were defined by values

25% above and below the baseline values. This was done so that shapes differing

much from the baseline shape are not attained and investigated during the

optimization process. Thus the required deformations from the baseline shape to the

optimized shape would be less.

Figure 30 B-Spline parameterization of the Boeing 737 airfoil (expanded)

105

The baseline, and the upper and lower limit values of the design variables are

as follows—

Table 6 Baseline and Upper and Lower limit Values of the Design Variables

DV CP Baseline Values Upper Limit Lower Limit 1 2 0.01189154138 0.014864426725 0.0089186560354

2 3 0.03237346633 0.040466832919 0.0242800997514

3 4 0.06064856760 0.075810709508 0.0454864257052

4 5 0.08333191253 0.104164890663 0.0624989343979 5 6 0.05126309035 0.064078862946 0.0384473177681

6 9 -0.03508193513 -0.043852418920 -0.0263114513520

7 10 -0.06910529131 -0.086381614144 -0.0518289684866

8 11 -0.04869035929 -0.060862949114 -0.0365177694684

9 12 -0.02601627424 -0.032520342809 -0.0195122056855

10 13 -0.00657293832 -0.008216172907 -0.0049297037445

11 Angle of Attack

2.5 to 3.5 1.5 4.0

4.3.4 CASE-A: Mach 0.74 at 9083.3 [m] (29,800[ft])

This corresponds to the off-design operating condition where the aircraft is

cruising in steady flight at Mach 0.74 (True airspeed) and 9083.33[m] (29800[ft]

above ground level). Following are the freestream conditions for this case—

Figure 31 Range of the Boeing-737 airfoil design variables

106

Table 7 Freestream conditions for the off-design operating point A

Following are the values of the optimum design variables for the operating

point-A—

Table 8 Baseline and Optimized values of the Design Variables

DV CP Baseline Values Optimum Values

1 2 0.01189154138 0.0094258191179

2 3 0.03237346633 0.0265314506846

3 4 0.06064856760 0.0716335747285

4 5 0.08333191253 0.066898113326

5 6 0.05126309035 0.0467090526596

6 9 -0.03508193513 -0.0283241637483

7 10 -0.06910529131 -0.0710298081602

8 11 -0.04869035929 -0.0395037365250

9 12 -0.02601627424 -0.023849412116

10 13 -0.00657293832 -0.0071745014079

11 Angle of Attack

2.627o 2.209o

The following values compare the performance parameters (Lift, moment and

drag) of the baseline and optimized airfoil. Here (C-W) refers to Clockwise moment

and (C-CW) refers to counter-clockwise moments—

True Airspeed Mach 0.74 [-]

224.4616 [m/s] (@9083.04[m])

Altitude 9083.3 [m] (29,800 [ft])

Atmospheric Pressure 30363.99 [N/m2] Air Density 0.461696 [kg/m3] Ambient Temperature 229.12024 [K] (-43.879 [oC])

Dynamic Viscosity 1.7894e-05 [kg/m-s] Reynolds Number 5.791x106 [-]

Angle of Attack 2.62712 o

107

Table 9 Baseline and Optimized performance at off-design operating point

PERFORMANCE PARAMETERS BASELINE OPTIMIZED REMARKS

LIFT FORCE Pressure Lift [N] 6141.1205 6153.514 Viscous Lift [N] 0.709057 0.09236 Total Lift [N] 6141.8296

[REQUIRED] 6153.6064

DIFFERENCE OF 0.19% (HIGHER)

Pressure CL [-] 0.5280 0.529069 Viscous CL [-] 6.0965e-05 7.94118e-06 Total CL [-] 0.52806

[REQUIRED] 0.529077

DIFFERENCE OF 0.19% (HIGHER)

MOMENTS Pressure Moment [N-m]

175.5258 150.5031

Viscous Moment [N-m]

0.15634 -0.41194

Total Moment [N-m]

175.6822 [C-CW] [REQUIRED]

150.09122 [C-CW]

DIFFERENCE OF 14.5% (LESS C-

CW) Pressure CM [-] -0.01509 0.01294 Viscous CM [-] 1.34421e-05 -3.5418e-05 Total CM [ -] 0.015104 [C-CW]

[REQUIRED] 0.012904 [C-

CW]

DIFFERENCE OF 14.5% (LESS C-

CW) DRAG FORCE Pressure Drag [N] 375.8002 169.5884 Viscous Drag [N] 55.4470 62.1306 Total Drag [N] 431.2472 231.71912

REDUCTION OF

46.26%

Pressure CD [-] 0.03231 0.01458 Viscous CD [-] 0.004767 0.0053419 Total CD [-] 0.0370779 0.0199228

REDUCTION OF

46.26%

∆ CD = -0.017155 ≈ 17 [DRAG COUNTS] LESS ANGLE OF ATTACK 2.6271o 2.209o

108

From the CP distribution it can be seen that the shock intensity has been

reduced and that supposedly the transition has been delayed, thus reducing shock

induced drag.

Model Evaluation

The RSM is constructed using 200 samples as in the DoE. Following this, 26

iterative improvements are performed on the RSM. Following are the RSM testing

values and a comparison between the actual values of the objective function and the

predicted values by the RSM—

Figure 32 CASE-A Baseline and Optimized airfoil shapes and their corresponding surface CP distributions

109

Fitness Parameters

- Root Mean Square Error (RMSE)

RMSE = 6.3575%

As the value of the RMSE is <10%, this indicates a reasonably good model

- Adjusted root mean square error σ𝑣

σ𝑣 = 0.6145

This is a small value compared to the data. Thus the model is good enough.

- Coefficient of multiple determination 𝑅𝑣𝑑𝑗2

𝑅𝑣𝑑𝑗2 = 0.9831

This value of 𝑅𝑣𝑑𝑗2 is close to 1. Thus it indicates a good model fit.

4.3.5 CASE-B: Mach 0.65 at 9083.3 [m] (29,800[ft])

This corresponds to the off-design operating condition where the aircraft is

cruising in steady flight at Mach 0.65(True airspeed) and 9083.33[m] (29800[ft]

above ground level). Following are the freestream conditions for this case—

Figure 33 Plot comparing the actual and the predicted objective function values (CASE-A)

110

Table 10 Freestream conditions for the off-design operating point B

Following are the values of the optimum design variables for the operating

point-A—

Table 11 Baseline and Optimized values of the Design Variables

DV CP Baseline Values Optimum Values

1 2 0.01189154138 0.010564514738021

2 3 0.03237346633 0.025643292357113

3 4 0.06064856760 0.072880447979943

4 5 0.08333191253 0.082824625968899

5 6 0.05126309035 0.042007873882886

6 9 -0.03508193513 -0.035352800149791

7 10 -0.06910529131 -0.058162753224287

8 11 -0.04869035929 -0.053120614103188

9 12 -0.02601627424 -0.030421540795948

10 13 -0.00657293832 -0.006672806068045

11 Angle of Attack

4.0271o 3.8795o

The following values compare the performance parameters (Lift, moment and

drag) of the baseline and optimized airfoil. Here (C-W) refers to Clockwise moment

and (C-CW) refers to counter-clockwise moments—

True Airspeed Mach 0.65 [-]

197.1622 [m/s] (@9083.04[m])

Altitude 9083.3 [m] (29,800 [ft])

Atmospheric Pressure 30363.99 [N/m2] Air Density 0.461696 [kg/m3] Ambient Temperature 229.12024 [K] (-43.879 [oC])

Dynamic Viscosity 1.7894e-05 [kg/m-s] Reynolds Number 5.087x106 [-]

Angle of Attack 4.0271 o

111

Table 12 Baseline and Optimized performance at off-design operating point B

PERFORMANCE PARAMETERS BASELINE OPTIMIZED REMARKS

LIFT FORCE Pressure Lift [N] 6141.1205 6189.2509 Viscous Lift [N] 0.709057 -0.06223 Total Lift [N] 6141.8296

[REQUIRED] 6189.1887 DIFFERENCE

OF 0.77% (HIGHER)

Pressure CL [-] 0.68434 0.689706 Viscous CL [-] 7.9015e-05 -6.935388e-06 Total CL [-] 0.68442

[REQUIRED] 0.6897 DIFFERENCE

OF 0.77% (HIGHER)

MOMENTS Pressure Moment [N-m] -39.8508 -35.6979 Viscous Moment [N-m]

-0.6048 -0.9138

Total Moment [N-m] -40.4556 [C-W] [REQUIRED]

-36.6117 [C-W] DIFFERENCE OF 9.5%

(LESS C-W) Pressure CM [-] -0.0044 -0.003978 Viscous CM [-] -6.73958e-05 -0.0001018 Total CM [-] -0.004508 [C-W]

[REQUIRED] -0.004079 [C-W] DIFFERENCE

OF 9.5% (LESS C-W)

DRAG FORCE Pressure Drag [N] 157.172 70.82073 Viscous Drag [N] 46.0286 50.04897 Total Drag [N] 203.2007 120.8697 REDUCTION

OF 40.5%

Pressure CD [-] 0.017515 0.0078919 Viscous CD [-] 0.005129 0.0055772 Total CD [-] 0.022643 0.0134692 REDUCTION

OF 40.5%

∆ CD = -0.009174 ≈ 9 [DRAG COUNTS] LESS ANGLE OF ATTACK 4.0271o 3.8795o

112

From the CP distribution it can be seen that the shock intensity has been

reduced the turbulence has not been delayed much, the reduction shock induced drag

seems to have contributed to the overall drag reduction significantly.

Model Evaluation

The RSM is constructed using 200 samples as in the DoE. Following this, 26

iterative improvements are performed on the RSM. Following are the RSM testing

values and a comparison between the actual values of the objective function and the

predicted values by the RSM—

Figure 34 CASE-B Baseline and Optimized airfoil shapes and their corresponding surface CP distributions

113

Fitness Parameters

- Root Mean Square Error (RMSE)

RMSE = 9.995%

As the value of the RMSE is <10%, this indicates a reasonably good model

- Adjusted root mean square error σ𝑣

σ𝑣 = 0.1235

This is a small value compared to the data. Thus the model is good enough.

- Coefficient of multiple determination 𝑅𝑣𝑑𝑗2

𝑅𝑣𝑑𝑗2 = 0.9885

This value of 𝑅𝑣𝑑𝑗2 is close to 1. Thus it indicates a good model fit.

4.3.6 CASE-C: Mach 0.65 at 6827.0 [m] (22,400[ft])

This corresponds to the off-design operating condition where the aircraft is

cruising in steady flight at Mach 0.65(True airspeed) and 6827[m] (22400[ft] above

ground level). Following are the freestream conditions for this case—

Figure 35 Plot comparing the actual and the predicted objective function values (CASE-B)

114

Table 13 Freestream conditions for the off-design operating point C

Following are the values of the optimum design variables for the operating

point-A—

Table 14 Baseline and Optimized values of the Design Variables C

DV CP Baseline Values Optimum Values

1 2 0.01189154138 0.0128883193059

2 3 0.03237346633 0.0275768561427

3 4 0.06064856760 0.0714721803511

4 5 0.08333191253 0.0689957016386

5 6 0.05126309035 0.0385565064030

6 9 -0.03508193513 -0.0283429759241

7 10 -0.06910529131 -0.0667506140691

8 11 -0.04869035929 -0.0503457197813

9 12 -0.02601627424 -0.0297665255502

10 13 -0.00657293832 -0.00786047883110

11 Angle of Attack

2.55o 2.7346884

The following values compare the performance parameters (Lift, moment and

drag) of the baseline and optimized airfoil. Here (C-W) refers to Clockwise moment

and (C-CW) refers to counter-clockwise moments—

True Airspeed Mach 0.65 [-]

203.45 [m/s] (@6827.0[m])

Altitude 6827.0 [m] (22,400 [ft])

Atmospheric Pressure 42067.06 [N/m2] Air Density 0.6012[kg/m3] Ambient Temperature 243.7811[K] (-29.368[oC])

Dynamic Viscosity 1.7894e-05 [kg/m-s] Reynolds Number 6.835x106 [-]

Angle of Attack 2.55o

115

Table 15 Baseline and Optimized performance at off-design operating point C

PERFORMANCE PARAMETERS BASELINE OPTIMIZED REMARKS

LIFT FORCE Pressure Lift [N] 6141.1205 6066.573 Viscous Lift [N] 0.709057 -0.56088 Total Lift [N] 6141.8296

[REQUIRED] 6066.01273 DIFFERENCE

OF 1.23% (LOWER)

Pressure CL [-] 0.4936 0.4879 Viscous CL [-] 5.698e-05 -4.511e-05 Total CL [-] 0.4936

[REQUIRED] 0.487917 DIFFERENCE

OF 1.23% (LOWER)

MOMENTS Pressure Moment [N-m] 57.92965 97.35203 Viscous Moment [N-m]

-1.1563 -0.99956

Total Moment [N-m] 56.7733 [C-CW] [REQUIRED]

96.3524 DIFFERENCE OF 69.71%

(MORE C-CW) Pressure CM [-] 0.004655 0.0078604 Viscous CM [-] -9.2931e-05 -8.039e-05 Total CM [-] 0.00456 [C-CW]

[REQUIRED] 0.00775 DIFFERENCE

OF 69.71% (MORE C-W)

DRAG FORCE Pressure Drag [N] 63.479 44.9669 Viscous Drag [N] 70.80876 72..6274 Total Drag [N] 134.28784 117.59436 REDUCTION

OF 12.43%

Pressure CD [-] 0.0051059 0.003616 Viscous CD [-] 0.0056954 0.005841 Total CD [-] 0.0108013 0.009458 REDUCTION

OF 12.43%

∆ CD =-1.345 ≈ 1.5 [DRAG COUNTS] LESS ANGLE OF ATTACK 2.55o 2.735o

116

From the CP distribution it can be seen that the shock intensity has drastically

been reduced when compared to the baseline airfoil. Thus the shock induced drag has

been reduced. However the lift generated is slightly less when compared to the

required lift.

Model Evaluation

The RSM is constructed using 200 samples as in the DoE. Following this, 26

iterative improvements are performed on the RSM. Following are the RSM testing

values and a comparison between the actual values of the objective function and the

predicted values by the RSM—

Figure 36 Plot comparing the actual and the predicted objective function values (CASE-C)

117

Fitness Parameters

- Adjusted root mean square error σ𝑣

σ𝑣 = 0.53410

This is a small value compared to the data. Thus the model is good enough.

- Coefficient of multiple determination 𝑅𝑣𝑑𝑗2

𝑅𝑣𝑑𝑗2 = 0.8873760

This value of 𝑅𝑣𝑑𝑗2 is close to 1. Thus it indicates a decent model fit.

4.4 Performance Comparison of Baseline Airfoil Relative to Optimized Airfoil

Additional flow simulations are performed to assess the performance

improvement (drag reduction) of the optimized airfoil relative to the baseline airfoil.

This is done on the basis of three criteria as follows—

- Fully optimized airfoil (Both optimized shape and angle of attack).

- Baseline airfoil shape at optimized angle of attack.

- Optimized airfoil shape at baseline angle of attack.

This analysis is done for the all the case studies discussed above— RAE 2822

Design study, Boeing 737C (Cases A, B and C). The change in drag is computed

relative to the baseline airfoil drag

Figure 37 Plot comparing the actual and the predicted objective function values (CASE-C)

118

Table 16 Performance Comparison Analysis

CASES Particulars Baseline Airfoil

Performance Comparison Criteria

Fully Optimized

Airfoil

Baseline Airfoil Shape @ Optimized

AoA

Optimized Airfoil Shape @ Baseline

AoA

RAE-2822 Design Study

DRAG [N] 270.9082 206.989 211.9949 281.4196 AoA 2.7 2.24 2.24 2.7 Drag

Change - -23.59% -21.74 +3.88%

Boeing 737C CASE-A M 0.74,

H 29800ft

DRAG [N] 431.2472 213.7191 360.9521 309.6035 AoA 2.627 2.209 2.209 2.627 Drag

Change - -46.26% -16.29% -28.2%

Boeing 737C CASE-B M 0.65,

H 29800 ft

DRAG [N] 203.2007 120.8697 187.8078 129.9117 AoA 4.027 3.879 3.879 4.027 Drag

Change - -40.5% -7.5% -36.06%

Boeing 737C CASE-C M 0.65,

H 22400 ft

DRAG [N] 134.2878 117.5943 144.0607 115.2182 AoA 2.55 2.735 2.735 2.55 Drag

Change - -12.43% +7.27% -14.2%

AoA Angle of Attack

4.5 Adaptive Airfoils

The adaptive airfoil is designed by optimizing the shape for the 3 cases (A, B

and C) as discussed above. It can be seen from the above results that the individual

single point optimized airfoils have less drag than the baseline drag. A maximum of

46% reduction is achieved using the adaptive airfoil for a particular operating point.

This is a significant drag reduction considering the amount of variation in the shape

required

Maximum movement of the upper surface is roughly 1.1% chord while the

maximum lower surface movement is roughly 0.3% chord. The maximum variation in

the lift force generated is 1.23%.

119

Note that the adaptive concept can be utilized in two different ways. When

operating at a given Mach number, the shape can be altered to match the airfoil

optimized at the Mach number closest to the given Mach number. Alternatively, one

can determine a continuous variation of shape throughout the Mach number range.

Figure 38 Baseline and Optimized Airfoil shapes for Cases A, B, and C

Figure 39 Baseline and Optimized Airfoil shapes for Cases A, B, and C (Expanded View)

120

5. CONCLUSION AND FUTURE WORK

5.1 Summary and Conclusion

In this research, an iterative response surface optimization methodology is

employed for carrying out aerodynamic shape optimization on various case studies.

These include a design validation study on the RAE 2822 airfoil and the shape

optimization of a Boeing-737 classic airfoil at various off-design operating

conditions.

The parameterization is carried out using B-spline control points, the y-

ordinates of which, and the angle of attack are taken as the design variables for the

optimization process. Using the results obtained from CFD simulations, a full

quadratic polynomial response surface model (RSM) is constructed, which is then

optimized using the SQP technique to obtain the optimum values of the design

variables. For constructing the RSM, the Latin Hypercube Sampling (LHS) design is

used to obtain the Design of Experiments (DoE) plan. Polynomial regression is used

to obtain the values of the regression coefficients. The RSM is improved in an

iterative manner, in that a CFD simulation is performed for the optimum values of the

design variables obtained at the end of each optimization step and the resulting value

of the objective function (actual value) along with the design variable values are

added to the initial DoE plan. Polynomial regression is performed again and new

values of the regression coefficients are obtained which are then used to re-construct

the RSM. This process is repeated till the difference between the actual and

approximated objective function value (value obtain from the RSM) does not lie

within a tolerance limit.

The main advantage of using this approach for shape optimization problems is

that values obtained from commercially available flow solvers can directly be used in

the optimization process, without making any changes to the solver’s code. Also the

noise and non-smoothness issues associated with CFD results are smoothened out by

using the RSM which is quadratic polynomial in terms of the design variables. Thus

the optimization process can be performed effectively and smoothly without any

sudden divergence issues associated with the CFD results.

The mesh required for the simulation is generated using GridPro which is an

automatic, topology based multi-block structured grid generator with an advanced

121

grid smoothening algorithm. ANSYS-FLUENT is used as the flow solver to perform

the CFD simulation. Here the governing flow equations, the continuity, Navier-Stokes

and the energy conservation equations are solved using a pressure-based solver. In

addition, the Spalart-Allmaras turbulence model, which is a one equation fully

turbulent model, is used to close the momentum equations. The results obtained from

the CFD simulations are the lift, drag and moment coefficients of the airfoil given the

free-stream conditions. These values are then used to evaluate the objective function

value.

The objective function used here is the lift-moment constrained drag

minimization problem. It is a penalty augmented type problem in which the

constraints (lift and moment coefficients) are added to the objective function as a

quadratic penalty term. Doing so converts the constrained problem into an

unconstrained problem and thus there is no requirement for the evaluation of the

constraints and their gradients separately.

MATLAB Optimization and Global Optimization Toolbox functions are used

to perform the optimization process. The main tasks performed using MATLAB are—

the parameterization of the airfoil shape, the DoE plan, RSM construction and the

iterative RSO processes. fmincon and MultiStart solvers are employed using the SQP

algorithm to optimize the RSM and obtain values of the optimum design variables.

As mentioned earlier, the iterative RSO methodology has been used here for

the aerodynamic shape optimization of various airfoils. These include—the RAE

2822 airfoil as a design validation study, and the Boeing-737 airfoil at 4 different off-

design operating conditions.

The Boeing 737 classic airfoil is optimized at four off-design operating

conditions. This has been used as a representative case-study to demonstrate the

advantages of using an adaptive airfoil over a fixed geometry airfoil. The results

obtained are encouraging in lieu of the fact that significant drag reduction is achieved

by morphing the airfoil shape from one operating condition to the other. The baseline

drag at off-design conditions is found to be significantly higher than that of the

optimized adaptive airfoil shape at the same condition. The drag-reduction is

attributed primarily to the reduction in the intensity of the shock wave on the upper

surface of the airfoil. Also the location of the adverse pressure gradient is pushed back

122

downstream (towards the trailing edge), thus delaying the transition from laminar and

turbulent. This is achieved by morphing the baseline geometry to a better suited one

which is obtained from the aerodynamic shape optimization process. The optimized

shape offers lesser drag while maintaining the desired lift force and the baseline

pitching moment. The idea behind retaining the baseline pitching moment is that, for

an optimized geometry, if there is a change in the pitching moment of the airfoil, the

aircraft elevators would have to be deployed to account for this change. This

indirectly would offset the benefit obtained from morphing the airfoil.

About 40% drag reduction is achieved by implementing the aerodynamic

shape optimization methodology that has been used in this research. The maximum

change in the geometry (from baseline to optimized) is about 0.2% of the airfoil

chord-length which is relatively very small compared to the overall dimensions of the

airfoil. This implies that very little deformations are required to obtain

aerodynamically efficient airfoils, which offer significantly lesser drag over a wider

portion of the flight envelope compared to fixed geometry airfoils which have been

optimized for only few operating points in the envelope.

Thus the adaptive airfoil concept is a viable concept which can pursued with

confidence. Given the fact that little deformations are required to obtain much

efficient airfoil shapes, the adaptive airfoil concept can be considered feasible from

the structural point of view as well.

However there are a lot of issues that remain to be addressed before this

technology can be implemented. Even though the required deformations are small,

there are structural challenges that hamper the full-scale development of this

technology. The ideal material that can be used as the skin for adaptive airfoils should

have high strength, can withstand huge strains without permanent deformation with

the ability to regain the original shape precisely, low stiffness, excellent fatigue

properties and low weight. Such a material is central to the development of this

concept. But as of now this area is still under-research. In addition to materials, robust

mechanisms have to be developed that can deform the skin to the desired (optimized)

shapes. Such mechanisms should be feasible, not involve a very complex architecture

and require very little actuation energy. A control system is then required that can

sense the change in the aircraft operating condition, and provides the necessary

signals that can actuate the mechanisms to obtain the desired shape.

123

5.2 Future Work

There remains a lot of scope for improvement in the optimization

methodology that has been employed in this research work. In particular the flow

solver and the turbulence model that has been used here can be replaced with higher-

fidelity flow solvers that can predict the flow transition from laminar to turbulent

more accurately. Laminar-turbulent flow transition prediction plays a vital role in the

accuracy of the results, particularly in drag prediction. However the most commonly

used turbulence model do not predict the transition well, and thus they can be

replaced by other models such as the k-kl-w Transition or the Transition SST models.

Also Large-Eddy Simulation (LES), in which the large scale eddies

(turbulence) are filtered and resolved, can be employed instead of the RANS

equations in which the eddies are time averaged. Thus more accurate results can be

obtained.

For the optimization of the response surface model, Genetic Algorithms or

Particle Swarm Optimization techniques can be investigated to obtain the optimum

values of the design variables, instead of the gradient based optimization method

(SQP) that has been used here. Using Genetic Algorithms and other similar

techniques can improve the chances of locating a global minimum instead of the local

minimum that is obtained using gradient based methods.

In this work, the methodology has been applied for the optimization of an

airfoil which is basically a 2-dimensional shape. Even though having the right airfoil

shape is essential, the 3-dimensional shape of the wing is equally crucial. A lot of

other factors need to be addressed while designing the 3-dimesnsional shape of the

wing such as tip vortices, engine and the fuselage interference effects etc. Thus for a

full-fledged implementation of the adaptive airfoil concept, it is necessary to extend

the optimization methodology to 3-dimensional wings. Here, with an addition of a

third dimension, more number of design variables will be required to completely

parameterize the wing.

The optimization objectives can also be extended to include other performance

parameters such as actuator energy requirements which are based on the energy

required to overcome inertia, material stiffness and the aerodynamic loads in order to

124

morph the shape. This would require fluid structure interaction analysis, a subject

that has drawn a lot of attention in the past few years.

The optimization methodology can be performed on more number of points

inside the flight envelope. This would ensure that a much wider area of the flight

envelope is covered, thus offering more operational flexibility while keeping the

overall drag to a minimum throughout the flight operation.

The results from this research show that significant drag reduction is possible

using the adaptive airfoil concept. This however must be weighed against the energy

consumed in adapting the wing, the increased cost and complexity, and the

implications for safety. As the actuator technology needed to deform the wing

improves, the price of fuel increases, and the need to reduce harmful emissions

becomes more critical, the adaptive wing concept will become increasingly viable.

125

REFERENCES

[1] Boeing. (2012, 13/05/2012). Boeing Long-Term Market Forecast summary.

Available: http://www.boeing.com/commercial/cmo/forecast_summary.html

[2] J. E. Hicken, "Efficient algorithms for future aircraft design: Contributions to

aerodynamic shape optimization," Doctor of Philosophy Doctoral thesis, Graduate

Department of Aerospace Science and Engineering, University of Toronto, Toronto,

Canada, 2009.

[3] Airbus. (2012). Biomimicry. Available: http://www.airbus.com/innovation/eco-

efficiency/design/biomimicry/

[4] D. Lentink, U. K. Müller, E. J. Stamhuis, R. de Kat, W. van Gestel, L. L. M.

Veldhuis, P. Henningsson, A. Hedenström, J. J. Videler, and J. L. van Leeuwen,

"How swifts control their glide performance with morphing wings," Nature, vol. 446,

pp. 1082-1085.

[5] T. A. Weisshaar, "Morphing Aircraft Technology - New Shapes for Aircraft Design,"

01 OCT 2006 2012.

[6] G. E. Lewis, R. E. Thomasson, and D. W. Nelson, "Wing Lift/Drag Optimizing

System," 1990.

[7] R. W. Wlezien, G. C. Homer, and A. R. McGowan, "THE AIRCRAFT MORPHING

PROGRAM," American Institute of Aeronautics and Astronautics, Long Beach,

California, USA1998.

[8] G. Dockter, "Geometric Morphing Wing with Exapandable Spars," United States of

America Patent, 2003.

[9] M. K. V. Sankrithi and J. B. Frommer, "Controllable Winglets," 2010.

[10] S. National Research Council . Division on Engineering and Physical and A. National

Research Council . Steering Committee for the Decadal Survey of Civil, Decadal

survey of civil aeronautics: foundation for the future. Washington, D.C: The National

Academies Press, 2006.

[11] E. Stanewsky, "Adaptive wing and flow control technology," Progress in Aerospace

Sciences, vol. 37, pp. 583-667, 2001.

[12] N. J. Pern, Jacob J. D., "Aerodynamic Flow Control Using Shape Adaptive Surfaces,"

in ASME Design Engineering Technical Conferences, Las Vegas, Nevada, USA,

1999.

126

[13] A. Jameson, N. A. Pierce, and L. Martinelli, "Optimum Aerodynamic Design using

the Navier-Stokes Equations," AIAA, 1997.

[14] A. Jameson, R. Hu, and A. C. Brown, "Aerodynamic Shape Optimization for the

World's Fastest P-51," presented at the 44th Aerospace Sciences Meeting and Exhibit,

Reno, Nevada, 2006.

[15] A. Jameson, "Aerodynamic Design via Control Theory," Journal of Scientific

Computing, vol. 3, 1988.

[16] A. Jameson, "Optimum Aerodynamic Design using CFD and Control Theory," AIAA,

1995.

[17] F. Austin, M. J. Rossi, W. van Nostrand, J. Su, G. Knowles, and A. Jameson, "Active

Rib Experiment for Adaptive Conformal Wing," in Third International Conference on

Adaptive Structures, 00/1993, p. 43.

[18] D. W. Zingg, L. Diosady, and L. Billing, "Adaptive Airfoils for Drag Reduction at

Transonic Speeds," presented at the 24th Applied Aerodynamic Conference, San

Francisco, California, 2006.

[19] H. Namgoong, W. A. Crossley, and A. S. Lyrintzis, "Aerodynamic Optimization of a

Morphing Airfoil Using Energy as an Objective," AIAA journal, vol. 45, pp. 2113-

2124, 2007.

[20] P. Gamboa, J. Vale, P. Lau, and A. Suleman, "Optimization of a Morphing Wing

Based on Coupled Aerodynamic and Structural Constraints," AIAA journal, vol. 47,

2009.

[21] N. M. Ursache, "The Design of Adaptive Structures for Wing Morphing," Ph.D,

Faculty of Engineering, Science and Mathematics, University of Southampton, 2006.

[22] L. Unlusoy, "Structural Design and Analysis of the Mission Adaptive Wings of an

Unmanned Aerial Vehicle," Master's, Graduate School of Natural and Applied

Sciences, Middle East Technical University, 2010.

[23] J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. New York, NY:

Springer Science+Business Media, LLC, 2006.

[24] S. S. Rao, Engineering Optimization - Theory and Practice 4ed.: John Wiley & Sons.

[25] E. Üstündag and M. S. Çelebi, "A B-Spline Curve Fitting Approach by Implementing

the Parameter Correction Terms."

[26] J. Hoschek, F. Schneider, and P. Wassum, "Optimal Approximate Conversion of

Spline Surfaces," Computer-Aided Design 6, pp. 293 -306, 1983.

127

[27] L. K. Billing, "On The Development Of An Improved Lift-Constrained Aerodynamic

Optimization Algorithm," Masters Thesis, Graduate Department of Aerospace

Engineering, University of Toronto, Toronto, Canada, 2006.

[28] J. D. Anderson, Fundamentals of aerodynamics. Boston, Mass: McGraw-Hill, 2001.

[29] J. H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics, 3rd ed.:

Springer, 2002.

[30] J. M. McDonough, "Introductory Lectures on Turbulence," ed. Kentucky, USA:

University of Kentucky, 2004.

[31] ANSYS FLUENT Theory Guide Release 13.0. USA: ANSYS, Inc., 2010.

[32] M. El-Sayed, T. Sun, and J. Berry, "Shape optimization with computational fluid

dynamics," Advances in Engineering Software, vol. 36, pp. 607-613, 2005.

[33] A. I. J. Forrester and A. J. Keane, "Recent advances in surrogate-based optimization,"

Progress in Aerospace Sciences, vol. 45, pp. 50-79, 2009.

[34] J. Ahn, H. J. Kim, D. H. Lee, and O. H. Rho, "Response surface method for airfoil

design in transonic flow," Journal of aircraft, vol. 38, pp. 231-238, 2001.

[35] V. N. e. a. Queipo, "Surrogate-based analysis and optimization," Progress in

Aerospace Sciences, vol. 41, 2005.

[36] Y. Mack and e. al, "Surrogate Model-Based Optimization Framework: A Case Study

in Aerospace Design," Studies in Computational Intelligence (SCI), vol. 51, pp. 323-

342, 2007.

[37] A. Forrester, A. Sobester, and A. Keane, Engineering Design via Surrogate

Modelling: A Practical Guide: Wiley, 2008.

[38] A. Vavalle and N. Qin, "Iterative response surface based optimization scheme for

transonic airfoil design," Journal of aircraft, vol. 44, pp. 365-376, 2007.

[39] MATLAB Optimization ToolboxTM 5 User's Guide Mathworks, 2010.

[40] MATLAB Global Optimization ToolBox-User's Guide Mathworks, 2010.

128

Appendix A GridPro Topology Input Language (TIL) File

(FINAL3.FRA)

SET DIMENSION 2

SET GRIDDEN 16

SET DISPLAY.SURF ON

COMPONENT main()

BEGIN

INPUT 1 surf(sOUT (0..7));

INPUT 2 corn(sIN (1:1..8),cIN (-129));

END

COMPONENT surf()

BEGIN

s 0 -linear "GPro_af1.dat" ; #TIL:1:0

s 1 -linear "GPro_af2.dat" ; #TIL:1:1

s 2 -linear "C_L.dat" ; #TIL:1:2

s 3 -linear "Back_end.dat" ; #TIL:1:3

s 4 -linear "Wake_line.dat" -O ; #TIL:1:4

s 5 -linear "trail_top.dat" -O ; #TIL:1:5

s 6 -linear "trail_down.dat" -O ; #TIL:1:6

s 7 -linear "af_lead.dat" -O ; #TIL:1:7

LABEL UnknownLabel_0= s(2);

LABEL UnknownLabel_1= s(3);

LABEL UnknownLabel_2= s(2);

LABEL UnknownLabel_3= s(3);

LABEL UnknownLabel_4= s(2);

LABEL UnknownLabel_5= s(3);

LABEL UnknownLabel_6= s(2);

LABEL UnknownLabel_7= s(3);

LABEL UnknownLabel_8= s(2);

LABEL UnknownLabel_9= s(3);

LABEL UnknownLabel_10= s(2);

LABEL UnknownLabel_11= s(3);

LABEL UnknownLabel_12= s(2);

LABEL UnknownLabel_13= s(3);

LABEL UnknownLabel_14= s(2);

129

LABEL UnknownLabel_15= s(3);

LABEL UnknownLabel_16= s(2);

LABEL UnknownLabel_17= s(3);

LABEL _009_user9= s(2);

LABEL _009_user9= s(3);

END

COMPONENT corn(sIN s[0..7],cIN c[0..128])

BEGIN

c 0 15 -20 0 -s s:3 s:2 -L c:0 -p 15.92 -15.27 0 ; #TIL:2:0

c 1 15 20 0 -s s:3 s:2 -L c:1 -p 15.55 16.25 0 ; #TIL:2:1

c 2 -15.379608 12.026067 0 -s s:2 -L c:2 -p -4.856 16.29 0 -g 1; #TIL:2:2

c 3 -14.208652 -13.075139 0 -s s:2 -L c:3 -p -16.01 -4.648 0 -g 1; #TIL:2:3

c 4 0.27747889 0.06994594 0 -s s:0 -L c:4 -p -4.754 5.103 0 -g 2033; #TIL:2:4

c 5 0.75113867 0.03877387 0 -s s:0 -L c:5 -p 5.094 5.147 0 -g 2033; #TIL:2:5

c 6 0.29804802 -0.05459776 0 -s s:1 -L c:6 -p -4.666 -4.536 0 ; #TIL:2:6

c 7 0.70580358 -0.03767442 0 -s s:1 -L c:7 -p 5.193 -4.439 0 ; #TIL:2:7

c 8 -0.53387156 1.1149678 0 -L c:8 -p -4.754 5.103 0 -g 1; #TIL:2:8

c 9 -0.49950319 -1.0935291 0 -L c:9 -p -4.666 -4.536 0 -g 1; #TIL:2:9

c 10 -18.427811 -5.2222994 0 -s s:2 -L c:10 -p -16.05 0.3627 0 -g 1; #TIL:2:10

c 11 -0.97522092 -0.46675736 0 -L c:11 -p -4.711 0.4427 0 -g 1; #TIL:2:11

c 12 0.01205125 -0.02173760 0 -s s:1 -L c:12 -p -4.711 0.4427 0 -g 33; #TIL:2:12

c 13 0.10163954 -0.04295817 0 -s s:1 -L c:13 12 6 -p -4.688 -2.158 0 -g 1;#TIL:2:13

g 13 12 40 ;

g 13 6 54 ;

c 14 -0.78115013 -0.78870751 0 -L c:14 11 9 -p -4.688 -2.158 0 -g 1;#TIL:2:14

c 15 -16.756227 -9.3594344 0 -s s:2 -L c:15 10 3 -p -16.03 -2.255 0 -g 1;#TIL:2:15

130

c 16 0.09806832 0.05570585 0 -s s:0 -L c:16 4 -p -4.733 2.837 0 -g 2033;#TIL:2:16

g 16 4 54 ;

c 17 -0.81260331 0.7956845 0 -L c:17 8 -p -4.733 2.837 0 -g 1;#TIL:2:17

c 18 -17.386961 8.2344006 0 -s s:2 -L c:18 2 -p -16.06 2.772 0 -g 1;#TIL:2:18

c 19 0.00715576 0.03070321 0 -s s:0 -L c:19 16 -p -4.719 1.323 0 -g 2033;#TIL:2:19

g 19 16 40 ;

c 20 -0.98964462 0.46095311 0 -L c:20 17 -p -4.719 1.323 0 -g 1;#TIL:2:20

c 21 -18.647731 4.6642091 0 -s s:2 -L c:21 18 -p -16.05 1.249 0 -g 1; #TIL:2:21

c 22 1.0000000000 0.0000000000 0 -s s:6 s:5 s:4 s:1 s:0 -L c:22 5 7 -p 5.133 1.325 0 -g 2041;#TIL:2:22

g 22 5 60 ;

g 22 7 60 ;

c 23 15 -0.0983066870 0 -s s:4 s:3 -L c:23 -p 15.72 1.395 0 -g 17; #TIL:2:23

c 24 0.4849936670 -0.0512852062 0 -s s:1 -L c:24 7 6 -p 1.077 -4.479 0 -g 1;#TIL:2:24

g 24 7 54 ;

g 24 6 54 ;

c 25 -0.15831072 -1.3593258 0 -L c:25 9 -p 1.077 -4.479 0 -g 1;#TIL:2:25

c 26 -10.54709 -16.671646 0 -s s:2 -L c:26 3 -p 1.185 -15.51 0 -g 1; #TIL:2:26

c 27 0.4931796914 0.0639134891 0 -s s:0 -L c:27 5 4 -p 0.1027 5.125 0 -g 2033;#TIL:2:27

g 27 5 54 ;

g 27 4 54 ;

c 28 -0.17851629 1.3853468 0 -L c:28 8 -p 0.1027 5.125 0 -g 1;#TIL:2:28

c 29 -11.830347 15.728489 0 -s s:2 -L c:29 2 -p -0.006178 16.28 0 -g 1; #TIL:2:29

c 30 0.0042123225 0.0347450144 0 -L c:30 19 -p -4.719 1.323 0 -g 1; #TIL:2:30

g 30 19 24 ;

c 31 0.0971142250 0.0606139721 0 -L c:31 16 30 -p -4.733 2.837 0 -g 1; #TIL:2:31

131

c 32 0.2773615447 0.0749445585 0 -L c:32 4 31 -p -4.754 5.103 0 -g 1; #TIL:2:32

c 33 0.4935498630 0.0688997675 0 -L c:33 27 32 -p 0.1027 5.125 0 -g 1; #TIL:2:33

c 34 0.7517889821 0.0437313968 0 -L c:34 5 33 -p 5.094 5.147 0 -g 1; #TIL:2:34

c 35 0.7062760733 -0.0426520439 0 -L c:35 7 -p 5.193 -4.439 0 -g 1; #TIL:2:35

c 36 0.4852184837 -0.0562801494 0 -L c:36 24 35 -p 1.077 -4.479 0 -g 1; #TIL:2:36

c 37 0.2979851125 -0.0595973593 0 -L c:37 6 36 -p -4.666 -4.536 0 -g 1; #TIL:2:37

c 38 0.1009855635 -0.0479152191 0 -L c:38 13 37 -p -4.688 -2.158 0 -g 1; #TIL:2:38

c 39 0.0095806800 -0.0260845820 0 -L c:39 12 38 -p -4.711 0.4427 0 -g 1;#TIL:2:39

c 40 1 20 0 -s s:5 s:2 -L c:40 -p 5.331 16.27 0 -g 1;#TIL:2:40

c 41 1 -20 0 -s s:6 s:2 -L c:41 -p 6.534 -15.42 0 -g 1; #TIL:2:41

c 42 -0.003 0 0 -s s:7 s:1 s:0 -L c:42 19 12 -p -4.715 0.8951 0 -g 33; #TIL:2:42

g 42 19 36 ;

g 42 12 36 ;

c 43 -0.008 0 0 -s s:7 -L c:43 30 39 42 -p -4.715 0.8951 0 -g 1;#TIL:2:43

c 44 -1.07 0 0 -s s:7 -L c:44 20 11 -p -4.715 0.8951 0 -g 1;#TIL:2:44

c 45 -19.1 0 0 -s s:7 s:2 -L c:45 21 10 -p -16.05 0.8182 0 -g 1;#TIL:2:45

c 46 0.8220226547 0.5791445375 0 -L c:46 34 -p 5.094 5.147 0 -g 1;#TIL:2:46

g 46 34 18 ;

c 47 0.7573058268 -0.5802354926 0 -L c:47 35 -p 5.193 -4.439 0 -g 1;#TIL:2:47

c 48 0.5032038178 -0.4558756045 0 -L c:48 36 25 47 -p 1.077 -4.479 0 -g 1;#TIL:2:48

g 48 25 8 ;

c 49 0.2942105293 -0.3595736126 0 -L c:49 37 9 48 -p -4.666 -4.536 0 -g 1;#TIL:2:49

c 50 0.0695948522 -0.2858534952 0 -L c:50 38 14 49 -p -4.688 -2.158 0 -g 1;#TIL:2:50

132

c 51 -0.0892421031 -0.1999640159 0 -L c:51 39 11 50 -p -4.711 0.4427 0 -g 1;#TIL:2:51

c 52 -0.208 0 0 -s s:7 -L c:52 43 44 51 -p -4.715 0.8951 0 -g 1;#TIL:2:52

c 53 -0.1135251564 0.1964171705 0 -L c:53 30 20 52 -p -4.719 1.323 0 -g 1;#TIL:2:53

c 54 0.0513175085 0.2962039990 0 -L c:54 31 17 53 -p -4.733 2.837 0 -g 1;#TIL:2:54

c 55 0.2703208312 0.3748619277 0 -L c:55 32 8 54 -p -4.754 5.103 0 -g 1;#TIL:2:55

c 56 0.5231635916 0.4678020451 0 -L c:56 33 28 55 46 -p 0.1027 5.125 0 -g 1;#TIL:2:56

c 57 15 4.75 0 -s s:3 -L c:57 -p 15.7 2.774 0 -g 1;#TIL:2:57

c 58 15 -4.75 0 -s s:3 -L c:58 -p 15.74 -0.1492 0 -g 1;#TIL:2:58

c 59 1 1.9 0 -s s:5 -L c:59 -p 5.142 2.025 0 -g 12582913;#TIL:2:59

c 60 1 -1.9 0 -s s:6 -L c:60 -p 5.191 0.6298 0 -g 1; #TIL:2:60

c 61 1 0.05 0 -s s:5 -L c:61 22 34 -p 5.133 1.342 0 -g 1;#TIL:2:61

c 62 15 0.9016933130 0 -s s:3 -L c:62 23 -p 15.72 1.429 0 -g 1;#TIL:2:62

c 63 1 -0.05 0 -s s:6 -L c:63 22 35 -p 5.135 1.303 0 -g 1;#TIL:2:63

c 64 15 -1.0983066870 0 -s s:3 -L c:64 23 -p 15.72 1.346 0 -g 1;#TIL:2:64

c 65 0.30973 1.6663 0 -L c:65 59 28 46 -p 2.868 3.424 0 -g 1;#TIL:2:65

c 66 -6.6558587 18.803912 0 -s s:2 -L c:66 40 29 -p 2.923 16.27 0 -g 1;#TIL:2:66

c 67 1 0.6500000000 0 -s s:5 -L c:67 61 59 46 -p 5.136 1.582 0 -g 1;#TIL:2:67

c 68 15 2.0116933130 0 -s s:3 -L c:68 62 57 -p 15.71 1.901 0 -g 1;#TIL:2:68

c 69 1 -0.6500000000 0 -s s:6 -L c:69 63 60 47 -p 5.156 1.047 0 -g 1;#TIL:2:69

c 70 15 -2.2083066870 0 -s s:3 -L c:70 64 58 -p 15.73 0.7777 0 -g 1;#TIL:2:70

c 71 0.32361616 -1.6274478 0 -L c:71 60 25 47 -p 3.62 -1.321 0 -g 1;#TIL:2:71

c 72 -5.1198025 -19.27377 0 -s s:2 -L c:72 41 26 -p 4.492 -15.45 0 -g 1;#TIL:2:72

133

c 73 1 4.3299106 0 -s s:5 -L c:73 59 -p 5.175 4.507 0 -g 98305;#TIL:2:73

g 73 59 8 ;

c 74 15 7 0 -s s:3 -L c:74 57 -p 15.67 5.122 0 -g 12289;#TIL:2:74

c 75 -0.58536495 3.8241013 0 -L c:75 65 73 -p 2.878 5.662 0 -g 98305;#TIL:2:75

c 76 -1.4907674 3.3199326 0 -L c:76 28 75 -p 0.08373 7.069 0 -g 1;#TIL:2:76

c 77 -2.3056719 2.6722406 0 -L c:77 8 76 -p -4.772 7.052 0 -g 1;#TIL:2:77

c 78 -2.9337173 1.9277671 0 -L c:78 17 77 -p -6.706 2.826 0 -g 393217;#TIL:2:78

c 79 -3.3924358 0.97611577 0 -L c:79 20 78 -p -6.693 1.31 0 -g 393217;#TIL:2:79

c 80 -3.57757 0 0 -s s:7 -L c:80 44 79 -p -6.69 0.8817 0 -g 393217;#TIL:2:80

c 81 -3.3839423 -1.1899742 0 -L c:81 11 80 -p -6.687 0.4288 0 -g 393217;#TIL:2:81

c 82 -2.9437786 -1.9421683 0 -L c:82 14 81 -p -6.664 -2.175 0 -g 393217;#TIL:2:82

c 83 -2.3087376 -2.6727836 0 -L c:83 9 82 -p -6.642 -4.556 0 -g 1;#TIL:2:83

c 84 -1.5067772 -3.2887937 0 -L c:84 25 83 -p 1.096 -6.401 0 -g 1;#TIL:2:84

c 85 -0.36163206 -3.884064 0 -L c:85 71 84 -p 3.772 -3.783 0 -g 1;#TIL:2:85

c 86 1 -4.2415517 0 -s s:6 -L c:86 60 85 -p 5.425 -2.167 0 -g 1572865;#TIL:2:86

c 87 15 -7 0 -s s:3 -L c:87 58 -p 15.77 -2.784 0 -g 1572865;#TIL:2:87

c 88 1 9.3 0 -s s:5 -L c:88 40 73 -p 5.276 12.11 0 -g 32769;#TIL:2:88

g 88 40 8 ;

g 88 73 8 ;

c 89 15 11.5 0 -s s:3 -L c:89 1 74 -p 15.59 12.31 0 -g 4097;#TIL:2:89

c 90 -2.1768669 8.5100887 0 -L c:90 66 88 75 -p 2.907 12.52 0 -g 32769;#TIL:2:90

c 91 -4.3865592 7.5781182 0 -L c:91 29 90 76 -p 0.02562 13.02 0 -g 1;#TIL:2:91

c 92 -6.2301001 6.0427238 0 -L c:92 2 91 77 -p -4.826 13.02 0 -g 1;#TIL:2:92

134

c 93 -7.5858052 4.2258478 0 -L c:93 18 92 78 -p -12.75 2.791 0 -g 131073;#TIL:2:93

c 94 -8.421654 2.1496 0 -L c:94 21 93 79 -p -12.74 1.271 0 -g 131073;#TIL:2:94

c 95 -8.65483 0 0 -s s:7 -L c:95 45 94 80 -p -12.74 0.8407 0 -g 131073;#TIL:2:95

c 96 -8.344819 -2.3600329 0 -L c:96 10 95 81 -p -12.74 0.3861 0 -g 131073;#TIL:2:96

c 97 -7.6475522 -4.1163208 0 -L c:97 15 96 82 -p -12.72 -2.227 0 -g 131073;#TIL:2:97

c 98 -6.3737558 -5.9080515 0 -L c:98 3 97 83 -p -12.7 -4.615 0 -g 1;#TIL:2:98

c 99 -4.301992 -7.6013844 0 -L c:99 26 98 84 -p 1.153 -12.29 0 -g 1;#TIL:2:99

c 100 -1.7742141 -8.7165805 0 -L c:100 72 99 85 -p 4.237 -11.32 0 -g 1;#TIL:2:100

c 101 1 -9.3 0 -s s:6 -L c:101 41 100 86 -p 6.142 -10.73 0 -g 2621441;#TIL:2:101

c 102 15 -11.5 0 -s s:3 -L c:102 0 87 -p 15.87 -10.85 0 -g 2621441;#TIL:2:102

c 103 6.3 20 0 -s s:2 -L c:103 40 -p 8.677 16.26 0 -g 1;#TIL:2:103

g 103 40 90 ;

c 104 6.3 10.2 0 -L c:104 88 103 -p 8.654 12.18 0 -g 4097;#TIL:2:104

c 105 6.20049 5.6128913 0 -L c:105 73 104 -p 8.612 4.708 0 -g 12289;#TIL:2:105

c 106 6.0137047 3.5208954 0 -L c:106 59 105 -p 8.598 2.27 0 -g 50331649;#TIL:2:106

c 107 5.74 1.2384014220 0 -L c:107 106 67 -p 8.598 1.686 0 -g 1;#TIL:2:107

c 108 5.6 0.5184014220 0 -L c:108 107 61 -p 8.599 1.37 0 -g 1;#TIL:2:108

c 109 5.5 -0.0315985780 0 -s s:4 -L c:109 108 22 -p 8.599 1.348 0 -g 17;#TIL:2:109

c 110 5.6 -0.5815985780 0 -L c:110 109 63 -p 8.6 1.317 0 -g 1;#TIL:2:110

c 111 5.74 -1.3015985780 0 -L c:111 110 69 -p 8.618 0.9588 0 -g 1;#TIL:2:111

c 112 5.9140989 -3.2509713 0 -L c:112 60 111 -p 8.645 0.3748 0 -g 1;#TIL:2:112

c 113 6.1106403 -5.5190698 0 -L c:113 86 112 -p 8.812 -2.369 0 -g 1572865;#TIL:2:113

135

c 114 6.3 -10.2 0 -L c:114 101 113 -p 9.326 -10.77 0 -g 2621441;#TIL:2:114

c 115 6.3 -20 0 -s s:2 -L c:115 41 114 -p 9.607 -15.37 0 -g 1;#TIL:2:115

c 116 11.2 20 0 -s s:2 -L c:116 103 1 -p 12.14 16.26 0 -g 1;#TIL:2:116

g 116 103 30 ;

g 116 1 30 ;

c 117 11.2 10.946 0 -L c:117 104 89 116 -p 12.15 12.25 0 -g 4097;#TIL:2:117

c 118 11.083306 6.5259738 0 -L c:118 105 74 117 -p 12.17 4.917 0 -g 12289;#TIL:2:118

c 119 10.976038 4.4195581 0 -L c:119 106 57 118 -p 12.18 2.524 0 -g 1;#TIL:2:119

c 120 10.865314 1.4825897003 0 -L c:120 107 68 119 -p 12.19 1.795 0 -g 1;#TIL:2:120

c 121 10.7 0.6825897003 0 -L c:121 108 62 120 -p 12.19 1.4 0 -g 1;#TIL:2:121

c 122 10.6 -0.0674102997 0 -s s:4 -L c:122 109 23 121 -p 12.19 1.372 0 -g 17;#TIL:2:122

c 123 10.7 -0.8174102997 0 -L c:123 110 64 122 -p 12.19 1.332 0 -g 1;#TIL:2:123

c 124 10.819722 -1.6174102997 0 -L c:124 111 70 123 -p 12.21 0.8675 0 -g 1;#TIL:2:124

c 125 10.935462 -4.0232149 0 -L c:125 112 58 124 -p 12.22 0.1105 0 -g 1;#TIL:2:125

c 126 10.989641 -6.2251315 0 -L c:126 113 87 125 -p 12.32 -2.578 0 -g 1572865;#TIL:2:126

c 127 11.2 -10.8775 0 -L c:127 114 102 126 -p 12.63 -10.81 0 -g 2621441;#TIL:2:127

c 128 11.2 -20 0 -s s:2 -L c:128 115 0 127 -p 12.79 -15.32 0 -g 1;#TIL:2:128

END

Schedule File (TIL)

File (FINAL3.SCH) step 1: -S 1000 -w

write -f FINAL3.tmp

MATLAB Run Commands >> !Ggrid FINAL3.fra

>> !clu FINAL3.tmp -s 1 0.00025 1.05

136

>> movefile('FINAL3.tmp.conn','FINAL3.tmp.tmp.conn');

>>!"C:\GridPro/az_mngr/outU_fluent.script.bat"

"C:/~/~/FINAL3.tmp.tmp" "C:/~/~/FLO_MESH.msh"

Appendix B ANSYS-FLUENT JOURNAL FILE

(FLUENT_RUN.JOU)

(cx-gui-do cx-activate-item "MenuBar*ReadSubMenu*Mesh...")

(cx-gui-do cx-set-text-entry "Select File*FilterText" "D:\MASTERS THESIS\CFD work\F I N A L - O P E R A T I O N\CFD SLAVE\*")

(cx-gui-do cx-activate-item "Select File*Apply")

(cx-gui-do cx-set-text-entry "Select File*Text" "FLO_MESH.msh")

(cx-gui-do cx-activate-item "Select File*OK")

/define/models/viscous/spalart-allmaras? yes

/define/materials/change-create air air yes ideal-gas no no yes sutherland three-coefficient-method 1.716e-05 273.11 110.56 no no no

/mesh/modify-zones/merge-zones 10 11 ()

/mesh/modify-zones/merge-zones 2 3 4 5 6 7 ()

(load "OP.var")

/define/boundary-conditions/pressure-far-field pressure-far-field-10 no 0 no M no T no M_x no M_y no n yes no 10

/define/operating-conditions/operating-pressure P

/report/reference-values/compute pressure-far-field pressure-far-field-10

/solve/set/p-v-coupling 24

(cx-gui-do cx-activate-item "NavigationPane*Frame1*PushButton12(Solution Methods)")

/solve/set/discretization-scheme/density 1

/solve/set/discretization-scheme/pressure 12

/solve/set/discretization-scheme/mom 1

/solve/set/discretization-scheme/nut 1

/solve/set/discretization-scheme/temperature 1

/solve/set/under-relaxation/nut 0.9

(cx-gui-do cx-activate-item "NavigationPane*Frame1*PushButton13(Solution Controls)")

(cx-gui-do cx-set-real-entry-list "Solution Controls*Frame1*Table1*Frame4(Explicit Relaxation Factors)*Table4(Explicit Relaxation Factors)*RealEntry1(Momentum)" '( 0.5))

137

(cx-gui-do cx-activate-item "Solution Controls*Frame1*Table1*Frame4(Explicit Relaxation Factors)*Table4(Explicit Relaxation Factors)*RealEntry1(Momentum)")

(cx-gui-do cx-set-real-entry-list "Solution Controls*Frame1*Table1*Frame4(Explicit Relaxation Factors)*Table4(Explicit Relaxation Factors)*RealEntry3(Pressure)" '( 0.5))

(cx-gui-do cx-activate-item "Solution Controls*Frame1*Table1*Frame4(Explicit Relaxation Factors)*Table4(Explicit Relaxation Factors)*RealEntry3(Pressure)")

/solve/monitors/residual/convergence-criteria 0.0000001 0.0001 0.0001 1e-06 0.001

/solve/monitors/force/drag-coefficient yes 8 9 () yes no yes 2 no D_x D_y

/solve/monitors/force/lift-coefficient yes 8 9 () yes no yes 3 no L_x L_y

/solve/monitors/force/moment-coefficient yes 8 9 () yes no yes 4 no 0.25 0 0 0 1

/solve/initialize/hyb-initialization

/solve/iterate 300

/report/forces/wall-forces yes D_x D_y yes "DRAG.csv"

/report/forces/wall-forces yes L_x L_y yes "LIFT.csv"

/report/forces/wall-moments yes 0.25 0 0 0 1 yes "MOMENT.csv"

/plot/plot yes "CP_TOP.csv" no no no pressure-coefficient y 1 0 wall-8 ()

/plot/plot yes "CP_LOW.csv" no no no pressure-coefficient y 1 0 wall-9 ()

/plot/plot yes "Mach_TOP.csv" no no no mach-number y 1 0 wall-8 ()

/plot/plot yes "Mach_LOW.csv" no no no mach-number y 1 0 wall-9 ()

/plot/plot yes "Stress_TOP.csv" no no no wall-shear y 1 0 wall-8 ()

/plot/plot yes "Stress_LOW.csv" no no no wall-shear y 1 0 wall-9 ()

exit

ok

138

VITA

Mohammed Taha Shafiq Khot was born on September 4, 1987, in the emirate

of Dubai, United Arab Emirates. He is a citizen of the republic of India. He attended

the Sharjah Indian School where he studied till grade 10. He then moved to India

where he attended high school at R. J. College, Mumbai in 2005. After high school,

he joined University of Mumbai—K. J. Somaiya College of Engineering where he

studied Mechanical Engineering, graduating with a Bachelor’s degree (in First Class)

in September 2009.

After graduating, Mr. Taha joined the American University of Sharjah where

he pursued a master’s program in Mechanical Engineering. He was awarded the

Master of Science in Mechanical Engineering degree in 2012.